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case pos
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
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a : α
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|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
·
|
right
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
·
|
Mathlib.Data.Set.Image.651_0.IJFiTzmYGOCpPSd
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s
|
Mathlib_Data_Set_Image
|
case pos.h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s : Set α
a : α
t : Set α
h : t ⊆ insert a s
hs : a ∈ t
⊢ ∃ x ⊆ s, insert a x = t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
|
refine' ⟨t \ {a}, _, _⟩
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
|
Mathlib.Data.Set.Image.651_0.IJFiTzmYGOCpPSd
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s
|
Mathlib_Data_Set_Image
|
case pos.h.refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s : Set α
a : α
t : Set α
h : t ⊆ insert a s
hs : a ∈ t
⊢ t \ {a} ⊆ s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
·
|
rw [diff_singleton_subset_iff]
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
·
|
Mathlib.Data.Set.Image.651_0.IJFiTzmYGOCpPSd
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s
|
Mathlib_Data_Set_Image
|
case pos.h.refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s : Set α
a : α
t : Set α
h : t ⊆ insert a s
hs : a ∈ t
⊢ t ⊆ insert a s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
|
assumption
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
|
Mathlib.Data.Set.Image.651_0.IJFiTzmYGOCpPSd
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s
|
Mathlib_Data_Set_Image
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case pos.h.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s : Set α
a : α
t : Set α
h : t ⊆ insert a s
hs : a ∈ t
⊢ insert a (t \ {a}) = t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
·
|
rw [insert_diff_singleton, insert_eq_of_mem hs]
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
·
|
Mathlib.Data.Set.Image.651_0.IJFiTzmYGOCpPSd
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s
|
Mathlib_Data_Set_Image
|
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s : Set α
a : α
t : Set α
h : t ⊆ insert a s
hs : a ∉ t
⊢ t ⊆ s ∨ ∃ x ⊆ s, insert a x = t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
·
|
left
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
·
|
Mathlib.Data.Set.Image.651_0.IJFiTzmYGOCpPSd
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s
|
Mathlib_Data_Set_Image
|
case neg.h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s : Set α
a : α
t : Set α
h : t ⊆ insert a s
hs : a ∉ t
⊢ t ⊆ s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
|
exact (subset_insert_iff_of_not_mem hs).mp h
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
|
Mathlib.Data.Set.Image.651_0.IJFiTzmYGOCpPSd
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s
|
Mathlib_Data_Set_Image
|
case h.mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s : Set α
a : α
t : Set α
⊢ (t ⊆ s ∨ ∃ x ⊆ s, insert a x = t) → t ⊆ insert a s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
·
|
rintro (h | ⟨s', h₁, rfl⟩)
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
·
|
Mathlib.Data.Set.Image.651_0.IJFiTzmYGOCpPSd
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s
|
Mathlib_Data_Set_Image
|
case h.mpr.inl
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s : Set α
a : α
t : Set α
h : t ⊆ s
⊢ t ⊆ insert a s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
·
|
exact subset_trans h (subset_insert a s)
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
·
|
Mathlib.Data.Set.Image.651_0.IJFiTzmYGOCpPSd
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s
|
Mathlib_Data_Set_Image
|
case h.mpr.inr.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s : Set α
a : α
s' : Set α
h₁ : s' ⊆ s
⊢ insert a s' ⊆ insert a s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
·
|
exact insert_subset_insert h₁
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
·
|
Mathlib.Data.Set.Image.651_0.IJFiTzmYGOCpPSd
|
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
p : α → Prop
⊢ (∀ a ∈ range f, p a) ↔ ∀ (i : ι), p (f i)
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by
|
simp
|
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by
|
Mathlib.Data.Set.Image.697_0.IJFiTzmYGOCpPSd
|
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i)
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
p : ↑(range f) → Prop
H : ∀ (i : ι), p { val := f i, property := (_ : f i ∈ range f) }
x✝ : ↑(range f)
y : α
i : ι
hi : f i = y
⊢ p { val := y, property := (_ : ∃ y_1, f y_1 = y) }
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
|
subst hi
|
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
|
Mathlib.Data.Set.Image.700_0.IJFiTzmYGOCpPSd
|
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
p : ↑(range f) → Prop
H : ∀ (i : ι), p { val := f i, property := (_ : f i ∈ range f) }
x✝ : ↑(range f)
i : ι
⊢ p { val := f i, property := (_ : ∃ y, f y = f i) }
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
|
apply H
|
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
|
Mathlib.Data.Set.Image.700_0.IJFiTzmYGOCpPSd
|
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
p : α → Prop
⊢ (∃ a ∈ range f, p a) ↔ ∃ i, p (f i)
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by
|
simp
|
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by
|
Mathlib.Data.Set.Image.707_0.IJFiTzmYGOCpPSd
|
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i)
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
p : α → Prop
⊢ (∃ a ∈ range f, p a) ↔ ∃ i, p (f i)
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
|
simpa only [exists_prop] using exists_range_iff
|
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
|
Mathlib.Data.Set.Image.710_0.IJFiTzmYGOCpPSd
|
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i)
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
p : ↑(range f) → Prop
x✝ : ∃ a, p a
a : α
i : ι
hi : f i = a
ha : p { val := a, property := (_ : ∃ y, f y = a) }
⊢ ∃ i, p { val := f i, property := (_ : f i ∈ range f) }
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
|
subst a
|
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
|
Mathlib.Data.Set.Image.714_0.IJFiTzmYGOCpPSd
|
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
p : ↑(range f) → Prop
x✝ : ∃ a, p a
i : ι
ha : p { val := f i, property := (_ : ∃ y, f y = f i) }
⊢ ∃ i, p { val := f i, property := (_ : f i ∈ range f) }
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
|
exact ⟨i, ha⟩
|
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
|
Mathlib.Data.Set.Image.714_0.IJFiTzmYGOCpPSd
|
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
⊢ f '' univ = range f
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
|
ext
|
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
|
Mathlib.Data.Set.Image.730_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f
|
Mathlib_Data_Set_Image
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
x✝ : β
⊢ x✝ ∈ f '' univ ↔ x✝ ∈ range f
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
|
simp [image, range]
|
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
|
Mathlib.Data.Set.Image.730_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set α
⊢ f '' s ⊆ range f
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
|
rw [← image_univ]
|
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
|
Mathlib.Data.Set.Image.736_0.IJFiTzmYGOCpPSd
|
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set α
⊢ f '' s ⊆ f '' univ
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ];
|
exact image_subset _ (subset_univ _)
|
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ];
|
Mathlib.Data.Set.Image.736_0.IJFiTzmYGOCpPSd
|
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
i : ℕ
⊢ i ∈ range Nat.succ → 0 < i
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
|
rintro ⟨n, rfl⟩
|
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
|
Mathlib.Data.Set.Image.744_0.IJFiTzmYGOCpPSd
|
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i
|
Mathlib_Data_Set_Image
|
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
n : ℕ
⊢ 0 < Nat.succ n
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
|
exact Nat.succ_pos n
|
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
|
Mathlib.Data.Set.Image.744_0.IJFiTzmYGOCpPSd
|
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → γ
g : β → γ
⊢ range f ⊆ range g ↔ ∃ h, f = g ∘ h
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
|
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
|
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
|
Mathlib.Data.Set.Image.766_0.IJFiTzmYGOCpPSd
|
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set β
⊢ range f = s ↔ (∀ (a : α), f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
|
rw [← range_subset_iff]
|
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
|
Mathlib.Data.Set.Image.770_0.IJFiTzmYGOCpPSd
|
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set β
⊢ range f = s ↔ (range fun a => f a) ⊆ s ∧ ∀ b ∈ s, ∃ a, f a = b
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
|
exact le_antisymm_iff
|
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
|
Mathlib.Data.Set.Image.770_0.IJFiTzmYGOCpPSd
|
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
g : β → γ
⊢ range (g ∘ f) ⊆ range g
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
|
rw [range_comp]
|
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
|
Mathlib.Data.Set.Image.776_0.IJFiTzmYGOCpPSd
|
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
g : β → γ
⊢ g '' range f ⊆ range g
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp];
|
apply image_subset_range
|
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp];
|
Mathlib.Data.Set.Image.776_0.IJFiTzmYGOCpPSd
|
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : ι → α
⊢ range f = ∅ ↔ IsEmpty ι
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
|
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
|
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
|
Mathlib.Data.Set.Image.788_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
⊢ f '' s ∪ f '' sᶜ = range f
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
|
rw [← image_union, ← image_univ, ← union_compl_self]
|
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
|
Mathlib.Data.Set.Image.800_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
x : α
⊢ insert (f x) (f '' {x}ᶜ) = range f
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
|
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
|
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
|
Mathlib.Data.Set.Image.805_0.IJFiTzmYGOCpPSd
|
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t✝ : Set α
f : α → β
t : Set β
x : β
x✝ : x ∈ t ∩ range f
hx : x ∈ t
y : α
h_eq : f y = x
⊢ y ∈ f ⁻¹' t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by
|
rw [preimage, mem_setOf, h_eq]
|
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by
|
Mathlib.Data.Set.Image.809_0.IJFiTzmYGOCpPSd
|
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t✝ : Set α
f : α → β
t : Set β
x : β
x✝ : x ∈ t ∩ range f
hx : x ∈ t
y : α
h_eq : f y = x
⊢ x ∈ t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq];
|
exact hx
|
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq];
|
Mathlib.Data.Set.Image.809_0.IJFiTzmYGOCpPSd
|
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set β
hs : s ⊆ range f
⊢ f '' (f ⁻¹' s) = s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by
|
rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
|
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by
|
Mathlib.Data.Set.Image.815_0.IJFiTzmYGOCpPSd
|
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set β
⊢ f '' (f ⁻¹' s) = s → s ⊆ range f
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
|
intro h
|
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
|
Mathlib.Data.Set.Image.819_0.IJFiTzmYGOCpPSd
|
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set β
h : f '' (f ⁻¹' s) = s
⊢ s ⊆ range f
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
|
rw [← h]
|
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
|
Mathlib.Data.Set.Image.819_0.IJFiTzmYGOCpPSd
|
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set β
h : f '' (f ⁻¹' s) = s
⊢ f '' (f ⁻¹' s) ⊆ range f
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
|
apply image_subset_range
|
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
|
Mathlib.Data.Set.Image.819_0.IJFiTzmYGOCpPSd
|
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
p : Set β → Prop
⊢ (∃ s ⊆ range f, p s) ↔ ∃ s, p (f '' s)
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
|
rw [← exists_range_iff, range_image]
|
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
|
Mathlib.Data.Set.Image.835_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s)
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
p : Set β → Prop
⊢ (∃ s ⊆ range f, p s) ↔ ∃ a ∈ 𝒫 range f, p a
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image];
|
rfl
|
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image];
|
Mathlib.Data.Set.Image.835_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s)
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
p : Set β → Prop
⊢ (∃ s, ∃ (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s)
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by
|
simp
|
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by
|
Mathlib.Data.Set.Image.841_0.IJFiTzmYGOCpPSd
|
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s)
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
p : Set β → Prop
⊢ (∀ s ⊆ range f, p s) ↔ ∀ (s : Set α), p (f '' s)
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
|
rw [← forall_range_iff, range_image]
|
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
|
Mathlib.Data.Set.Image.845_0.IJFiTzmYGOCpPSd
|
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s)
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
p : Set β → Prop
⊢ (∀ s ⊆ range f, p s) ↔ ∀ a ∈ 𝒫 range f, p a
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image];
|
rfl
|
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image];
|
Mathlib.Data.Set.Image.845_0.IJFiTzmYGOCpPSd
|
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s)
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
⊢ f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
|
constructor
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
|
Mathlib.Data.Set.Image.849_0.IJFiTzmYGOCpPSd
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t
|
Mathlib_Data_Set_Image
|
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
⊢ f ⁻¹' s ⊆ f ⁻¹' t → s ⊆ t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
·
|
intro h x hx
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
·
|
Mathlib.Data.Set.Image.849_0.IJFiTzmYGOCpPSd
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t
|
Mathlib_Data_Set_Image
|
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
h : f ⁻¹' s ⊆ f ⁻¹' t
x : α
hx : x ∈ s
⊢ x ∈ t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
|
rcases hs hx with ⟨y, rfl⟩
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
|
Mathlib.Data.Set.Image.849_0.IJFiTzmYGOCpPSd
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t
|
Mathlib_Data_Set_Image
|
case mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
h : f ⁻¹' s ⊆ f ⁻¹' t
y : β
hx : f y ∈ s
⊢ f y ∈ t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
|
exact h hx
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
|
Mathlib.Data.Set.Image.849_0.IJFiTzmYGOCpPSd
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t
|
Mathlib_Data_Set_Image
|
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
⊢ s ⊆ t → f ⁻¹' s ⊆ f ⁻¹' t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
|
intro h x
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
|
Mathlib.Data.Set.Image.849_0.IJFiTzmYGOCpPSd
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t
|
Mathlib_Data_Set_Image
|
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
h : s ⊆ t
x : β
⊢ x ∈ f ⁻¹' s → x ∈ f ⁻¹' t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x;
|
apply h
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x;
|
Mathlib.Data.Set.Image.849_0.IJFiTzmYGOCpPSd
|
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
ht : t ⊆ range f
⊢ f ⁻¹' s = f ⁻¹' t ↔ s = t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
|
constructor
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
|
Mathlib.Data.Set.Image.858_0.IJFiTzmYGOCpPSd
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t
|
Mathlib_Data_Set_Image
|
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
ht : t ⊆ range f
⊢ f ⁻¹' s = f ⁻¹' t → s = t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
·
|
intro h
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
·
|
Mathlib.Data.Set.Image.858_0.IJFiTzmYGOCpPSd
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t
|
Mathlib_Data_Set_Image
|
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
ht : t ⊆ range f
h : f ⁻¹' s = f ⁻¹' t
⊢ s = t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
|
apply Subset.antisymm
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
|
Mathlib.Data.Set.Image.858_0.IJFiTzmYGOCpPSd
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t
|
Mathlib_Data_Set_Image
|
case mp.h₁
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
ht : t ⊆ range f
h : f ⁻¹' s = f ⁻¹' t
⊢ s ⊆ t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
·
|
rw [← preimage_subset_preimage_iff hs, h]
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
·
|
Mathlib.Data.Set.Image.858_0.IJFiTzmYGOCpPSd
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t
|
Mathlib_Data_Set_Image
|
case mp.h₂
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
ht : t ⊆ range f
h : f ⁻¹' s = f ⁻¹' t
⊢ t ⊆ s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
·
|
rw [← preimage_subset_preimage_iff ht, h]
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
·
|
Mathlib.Data.Set.Image.858_0.IJFiTzmYGOCpPSd
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t
|
Mathlib_Data_Set_Image
|
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
ht : t ⊆ range f
⊢ s = t → f ⁻¹' s = f ⁻¹' t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
|
rintro rfl
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
|
Mathlib.Data.Set.Image.858_0.IJFiTzmYGOCpPSd
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t
|
Mathlib_Data_Set_Image
|
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t s : Set α
f : β → α
hs ht : s ⊆ range f
⊢ f ⁻¹' s = f ⁻¹' s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl;
|
rfl
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl;
|
Mathlib.Data.Set.Image.858_0.IJFiTzmYGOCpPSd
|
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set β
⊢ f ⁻¹' (range f ∩ s) = f ⁻¹' s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
|
rw [inter_comm, preimage_inter_range]
|
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
|
Mathlib.Data.Set.Image.876_0.IJFiTzmYGOCpPSd
|
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set β
⊢ f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
|
rw [image_preimage_eq_inter_range, preimage_inter_range]
|
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
|
Mathlib.Data.Set.Image.880_0.IJFiTzmYGOCpPSd
|
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
⊢ range Sum.inl = {x | Sum.isLeft x = true}
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by
|
ext (_|_)
|
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by
|
Mathlib.Data.Set.Image.910_0.IJFiTzmYGOCpPSd
|
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x}
|
Mathlib_Data_Set_Image
|
case h.inl
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
val✝ : α
⊢ Sum.inl val✝ ∈ range Sum.inl ↔ Sum.inl val✝ ∈ {x | Sum.isLeft x = true}
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;>
|
simp
|
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;>
|
Mathlib.Data.Set.Image.910_0.IJFiTzmYGOCpPSd
|
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x}
|
Mathlib_Data_Set_Image
|
case h.inr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
val✝ : β
⊢ Sum.inr val✝ ∈ range Sum.inl ↔ Sum.inr val✝ ∈ {x | Sum.isLeft x = true}
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;>
|
simp
|
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;>
|
Mathlib.Data.Set.Image.910_0.IJFiTzmYGOCpPSd
|
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x}
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
⊢ range Sum.inr = {x | Sum.isRight x = true}
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by
|
ext (_|_)
|
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by
|
Mathlib.Data.Set.Image.912_0.IJFiTzmYGOCpPSd
|
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x}
|
Mathlib_Data_Set_Image
|
case h.inl
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
val✝ : α
⊢ Sum.inl val✝ ∈ range Sum.inr ↔ Sum.inl val✝ ∈ {x | Sum.isRight x = true}
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;>
|
simp
|
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;>
|
Mathlib.Data.Set.Image.912_0.IJFiTzmYGOCpPSd
|
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x}
|
Mathlib_Data_Set_Image
|
case h.inr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
val✝ : β
⊢ Sum.inr val✝ ∈ range Sum.inr ↔ Sum.inr val✝ ∈ {x | Sum.isRight x = true}
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;>
|
simp
|
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;>
|
Mathlib.Data.Set.Image.912_0.IJFiTzmYGOCpPSd
|
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x}
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
⊢ range Sum.inl ⊓ range Sum.inr ≤ ⊥
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
|
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
|
Mathlib.Data.Set.Image.915_0.IJFiTzmYGOCpPSd
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr)
|
Mathlib_Data_Set_Image
|
case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
x₁ : α
x₂ : β
h : Sum.inr x₂ = Sum.inl x₁
⊢ Sum.inl x₁ ∈ ⊥
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
|
exact Sum.noConfusion h
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
|
Mathlib.Data.Set.Image.915_0.IJFiTzmYGOCpPSd
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr)
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
⊢ ⊤ ≤ range Sum.inl ⊔ range Sum.inr
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by
|
rintro (x | y) - <;> [left; right]
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by
|
Mathlib.Data.Set.Image.915_0.IJFiTzmYGOCpPSd
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr)
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
⊢ ⊤ ≤ range Sum.inl ⊔ range Sum.inr
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by
|
rintro (x | y) -
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by
|
Mathlib.Data.Set.Image.915_0.IJFiTzmYGOCpPSd
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr)
|
Mathlib_Data_Set_Image
|
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
x : α
⊢ Sum.inl x ∈ range Sum.inl ⊔ range Sum.inr
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [
|
left
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [
|
Mathlib.Data.Set.Image.915_0.IJFiTzmYGOCpPSd
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr)
|
Mathlib_Data_Set_Image
|
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
y : β
⊢ Sum.inr y ∈ range Sum.inl ⊔ range Sum.inr
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left;
|
right
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left;
|
Mathlib.Data.Set.Image.915_0.IJFiTzmYGOCpPSd
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr)
|
Mathlib_Data_Set_Image
|
case inl.h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
x : α
⊢ Sum.inl x ∈ range Sum.inl
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;>
|
exact mem_range_self _
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;>
|
Mathlib.Data.Set.Image.915_0.IJFiTzmYGOCpPSd
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr)
|
Mathlib_Data_Set_Image
|
case inr.h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
y : β
⊢ Sum.inr y ∈ range Sum.inr
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;>
|
exact mem_range_self _
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;>
|
Mathlib.Data.Set.Image.915_0.IJFiTzmYGOCpPSd
|
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr)
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s✝ t : Set α
s : Set β
⊢ Sum.inl ⁻¹' (Sum.inr '' s) = ∅
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
|
ext
|
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
|
Mathlib.Data.Set.Image.943_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅
|
Mathlib_Data_Set_Image
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s✝ t : Set α
s : Set β
x✝ : α
⊢ x✝ ∈ Sum.inl ⁻¹' (Sum.inr '' s) ↔ x✝ ∈ ∅
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
|
simp
|
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
|
Mathlib.Data.Set.Image.943_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s✝ t s : Set α
⊢ Sum.inr ⁻¹' (Sum.inl '' s) = ∅
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
|
ext
|
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
|
Mathlib.Data.Set.Image.949_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅
|
Mathlib_Data_Set_Image
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s✝ t s : Set α
x✝ : β
⊢ x✝ ∈ Sum.inr ⁻¹' (Sum.inl '' s) ↔ x✝ ∈ ∅
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
|
simp
|
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
|
Mathlib.Data.Set.Image.949_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
⊢ Sum.inl ⁻¹' range Sum.inr = ∅
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
|
rw [← image_univ, preimage_inl_image_inr]
|
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
|
Mathlib.Data.Set.Image.955_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
⊢ Sum.inr ⁻¹' range Sum.inl = ∅
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
|
rw [← image_univ, preimage_inr_image_inl]
|
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
|
Mathlib.Data.Set.Image.960_0.IJFiTzmYGOCpPSd
|
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s✝ t : Set α
s : Set (α ⊕ β)
⊢ Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
|
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
|
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
|
Mathlib.Data.Set.Image.975_0.IJFiTzmYGOCpPSd
|
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
⊢ range (Subtype.map f h) = Subtype.val ⁻¹' (f '' {x | p x})
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
|
ext ⟨x, hx⟩
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
case h.mk
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
x : β
hx : q x
⊢ { val := x, property := hx } ∈ range (Subtype.map f h) ↔
{ val := x, property := hx } ∈ Subtype.val ⁻¹' (f '' {x | p x})
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
|
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
case h.mk
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
x : β
hx : q x
⊢ (∃ a, ∃ (b : p a), Subtype.map f h { val := a, property := b } = { val := x, property := hx }) ↔
∃ x_1 ∈ {x | p x}, f x_1 = x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
|
apply Iff.intro
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
case h.mk.mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
x : β
hx : q x
⊢ (∃ a, ∃ (b : p a), Subtype.map f h { val := a, property := b } = { val := x, property := hx }) →
∃ x_1 ∈ {x | p x}, f x_1 = x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
·
|
rintro ⟨a, b, hab⟩
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
·
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
case h.mk.mp.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
x : β
hx : q x
a : α
b : p a
hab : Subtype.map f h { val := a, property := b } = { val := x, property := hx }
⊢ ∃ x_1 ∈ {x | p x}, f x_1 = x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
|
rw [Subtype.map, Subtype.mk.injEq] at hab
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
case h.mk.mp.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
x : β
hx : q x
a : α
b : p a
hab : f ↑{ val := a, property := b } = x
⊢ ∃ x_1 ∈ {x | p x}, f x_1 = x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
|
use a
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
x : β
hx : q x
a : α
b : p a
hab : f ↑{ val := a, property := b } = x
⊢ a ∈ {x | p x} ∧ f a = x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
|
trivial
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
case h.mk.mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
x : β
hx : q x
⊢ (∃ x_1 ∈ {x | p x}, f x_1 = x) →
∃ a, ∃ (b : p a), Subtype.map f h { val := a, property := b } = { val := x, property := hx }
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
·
|
rintro ⟨a, b, hab⟩
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
·
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
case h.mk.mpr.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
x : β
hx : q x
a : α
b : a ∈ {x | p x}
hab : f a = x
⊢ ∃ a, ∃ (b : p a), Subtype.map f h { val := a, property := b } = { val := x, property := hx }
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
|
use a
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
x : β
hx : q x
a : α
b : a ∈ {x | p x}
hab : f a = x
⊢ ∃ (b : p a), Subtype.map f h { val := a, property := b } = { val := x, property := hx }
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
|
use b
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
x : β
hx : q x
a : α
b : a ∈ {x | p x}
hab : f a = x
⊢ Subtype.map f h { val := a, property := b } = { val := x, property := hx }
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
|
rw [Subtype.map, Subtype.mk.injEq]
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
q : β → Prop
f : α → β
h : ∀ (x : α), p x → q (f x)
x : β
hx : q x
a : α
b : a ∈ {x | p x}
hab : f a = x
⊢ f ↑{ val := a, property := b } = x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
|
exact hab
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
|
Mathlib.Data.Set.Image.1034_0.IJFiTzmYGOCpPSd
|
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x })
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : ι → α
x : α
⊢ range f ⊆ {x} ↔ f = const ι x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
exact hab
-- Porting note: `simp_rw` fails here
-- simp_rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.map, Subtype.coe_mk,
-- mem_set_of, exists_prop]
#align set.range_subtype_map Set.range_subtype_map
theorem image_swap_eq_preimage_swap : image (@Prod.swap α β) = preimage Prod.swap :=
image_eq_preimage_of_inverse Prod.swap_leftInverse Prod.swap_rightInverse
#align set.image_swap_eq_preimage_swap Set.image_swap_eq_preimage_swap
theorem preimage_singleton_nonempty {f : α → β} {y : β} : (f ⁻¹' {y}).Nonempty ↔ y ∈ range f :=
Iff.rfl
#align set.preimage_singleton_nonempty Set.preimage_singleton_nonempty
theorem preimage_singleton_eq_empty {f : α → β} {y : β} : f ⁻¹' {y} = ∅ ↔ y ∉ range f :=
not_nonempty_iff_eq_empty.symm.trans preimage_singleton_nonempty.not
#align set.preimage_singleton_eq_empty Set.preimage_singleton_eq_empty
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x := by
|
simp [range_subset_iff, funext_iff, mem_singleton]
|
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x := by
|
Mathlib.Data.Set.Image.1065_0.IJFiTzmYGOCpPSd
|
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set β
⊢ f '' (f ⁻¹' s)ᶜ = range f \ s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
exact hab
-- Porting note: `simp_rw` fails here
-- simp_rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.map, Subtype.coe_mk,
-- mem_set_of, exists_prop]
#align set.range_subtype_map Set.range_subtype_map
theorem image_swap_eq_preimage_swap : image (@Prod.swap α β) = preimage Prod.swap :=
image_eq_preimage_of_inverse Prod.swap_leftInverse Prod.swap_rightInverse
#align set.image_swap_eq_preimage_swap Set.image_swap_eq_preimage_swap
theorem preimage_singleton_nonempty {f : α → β} {y : β} : (f ⁻¹' {y}).Nonempty ↔ y ∈ range f :=
Iff.rfl
#align set.preimage_singleton_nonempty Set.preimage_singleton_nonempty
theorem preimage_singleton_eq_empty {f : α → β} {y : β} : f ⁻¹' {y} = ∅ ↔ y ∉ range f :=
not_nonempty_iff_eq_empty.symm.trans preimage_singleton_nonempty.not
#align set.preimage_singleton_eq_empty Set.preimage_singleton_eq_empty
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x := by
simp [range_subset_iff, funext_iff, mem_singleton]
#align set.range_subset_singleton Set.range_subset_singleton
theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s := by
|
rw [compl_eq_univ_diff, image_diff_preimage, image_univ]
|
theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s := by
|
Mathlib.Data.Set.Image.1069_0.IJFiTzmYGOCpPSd
|
theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set α
⊢ f '' s = range fun x => f ↑x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
exact hab
-- Porting note: `simp_rw` fails here
-- simp_rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.map, Subtype.coe_mk,
-- mem_set_of, exists_prop]
#align set.range_subtype_map Set.range_subtype_map
theorem image_swap_eq_preimage_swap : image (@Prod.swap α β) = preimage Prod.swap :=
image_eq_preimage_of_inverse Prod.swap_leftInverse Prod.swap_rightInverse
#align set.image_swap_eq_preimage_swap Set.image_swap_eq_preimage_swap
theorem preimage_singleton_nonempty {f : α → β} {y : β} : (f ⁻¹' {y}).Nonempty ↔ y ∈ range f :=
Iff.rfl
#align set.preimage_singleton_nonempty Set.preimage_singleton_nonempty
theorem preimage_singleton_eq_empty {f : α → β} {y : β} : f ⁻¹' {y} = ∅ ↔ y ∉ range f :=
not_nonempty_iff_eq_empty.symm.trans preimage_singleton_nonempty.not
#align set.preimage_singleton_eq_empty Set.preimage_singleton_eq_empty
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x := by
simp [range_subset_iff, funext_iff, mem_singleton]
#align set.range_subset_singleton Set.range_subset_singleton
theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s := by
rw [compl_eq_univ_diff, image_diff_preimage, image_univ]
#align set.image_compl_preimage Set.image_compl_preimage
/-- Any map `f : ι → β` factors through a map `rangeFactorization f : ι → range f`. -/
def rangeFactorization (f : ι → β) : ι → range f := fun i => ⟨f i, mem_range_self i⟩
#align set.range_factorization Set.rangeFactorization
theorem rangeFactorization_eq {f : ι → β} : Subtype.val ∘ rangeFactorization f = f :=
funext fun _ => rfl
#align set.range_factorization_eq Set.rangeFactorization_eq
@[simp]
theorem rangeFactorization_coe (f : ι → β) (a : ι) : (rangeFactorization f a : β) = f a :=
rfl
#align set.range_factorization_coe Set.rangeFactorization_coe
@[simp]
theorem coe_comp_rangeFactorization (f : ι → β) : (↑) ∘ rangeFactorization f = f := rfl
#align set.coe_comp_range_factorization Set.coe_comp_rangeFactorization
theorem surjective_onto_range : Surjective (rangeFactorization f) := fun ⟨_, ⟨i, rfl⟩⟩ => ⟨i, rfl⟩
#align set.surjective_onto_range Set.surjective_onto_range
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
|
ext
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
|
Mathlib.Data.Set.Image.1093_0.IJFiTzmYGOCpPSd
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x
|
Mathlib_Data_Set_Image
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set α
x✝ : β
⊢ x✝ ∈ f '' s ↔ x✝ ∈ range fun x => f ↑x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
exact hab
-- Porting note: `simp_rw` fails here
-- simp_rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.map, Subtype.coe_mk,
-- mem_set_of, exists_prop]
#align set.range_subtype_map Set.range_subtype_map
theorem image_swap_eq_preimage_swap : image (@Prod.swap α β) = preimage Prod.swap :=
image_eq_preimage_of_inverse Prod.swap_leftInverse Prod.swap_rightInverse
#align set.image_swap_eq_preimage_swap Set.image_swap_eq_preimage_swap
theorem preimage_singleton_nonempty {f : α → β} {y : β} : (f ⁻¹' {y}).Nonempty ↔ y ∈ range f :=
Iff.rfl
#align set.preimage_singleton_nonempty Set.preimage_singleton_nonempty
theorem preimage_singleton_eq_empty {f : α → β} {y : β} : f ⁻¹' {y} = ∅ ↔ y ∉ range f :=
not_nonempty_iff_eq_empty.symm.trans preimage_singleton_nonempty.not
#align set.preimage_singleton_eq_empty Set.preimage_singleton_eq_empty
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x := by
simp [range_subset_iff, funext_iff, mem_singleton]
#align set.range_subset_singleton Set.range_subset_singleton
theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s := by
rw [compl_eq_univ_diff, image_diff_preimage, image_univ]
#align set.image_compl_preimage Set.image_compl_preimage
/-- Any map `f : ι → β` factors through a map `rangeFactorization f : ι → range f`. -/
def rangeFactorization (f : ι → β) : ι → range f := fun i => ⟨f i, mem_range_self i⟩
#align set.range_factorization Set.rangeFactorization
theorem rangeFactorization_eq {f : ι → β} : Subtype.val ∘ rangeFactorization f = f :=
funext fun _ => rfl
#align set.range_factorization_eq Set.rangeFactorization_eq
@[simp]
theorem rangeFactorization_coe (f : ι → β) (a : ι) : (rangeFactorization f a : β) = f a :=
rfl
#align set.range_factorization_coe Set.rangeFactorization_coe
@[simp]
theorem coe_comp_rangeFactorization (f : ι → β) : (↑) ∘ rangeFactorization f = f := rfl
#align set.coe_comp_range_factorization Set.coe_comp_rangeFactorization
theorem surjective_onto_range : Surjective (rangeFactorization f) := fun ⟨_, ⟨i, rfl⟩⟩ => ⟨i, rfl⟩
#align set.surjective_onto_range Set.surjective_onto_range
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
|
constructor
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
|
Mathlib.Data.Set.Image.1093_0.IJFiTzmYGOCpPSd
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x
|
Mathlib_Data_Set_Image
|
case h.mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set α
x✝ : β
⊢ x✝ ∈ f '' s → x✝ ∈ range fun x => f ↑x
case h.mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set α
x✝ : β
⊢ (x✝ ∈ range fun x => f ↑x) → x✝ ∈ f '' s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
exact hab
-- Porting note: `simp_rw` fails here
-- simp_rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.map, Subtype.coe_mk,
-- mem_set_of, exists_prop]
#align set.range_subtype_map Set.range_subtype_map
theorem image_swap_eq_preimage_swap : image (@Prod.swap α β) = preimage Prod.swap :=
image_eq_preimage_of_inverse Prod.swap_leftInverse Prod.swap_rightInverse
#align set.image_swap_eq_preimage_swap Set.image_swap_eq_preimage_swap
theorem preimage_singleton_nonempty {f : α → β} {y : β} : (f ⁻¹' {y}).Nonempty ↔ y ∈ range f :=
Iff.rfl
#align set.preimage_singleton_nonempty Set.preimage_singleton_nonempty
theorem preimage_singleton_eq_empty {f : α → β} {y : β} : f ⁻¹' {y} = ∅ ↔ y ∉ range f :=
not_nonempty_iff_eq_empty.symm.trans preimage_singleton_nonempty.not
#align set.preimage_singleton_eq_empty Set.preimage_singleton_eq_empty
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x := by
simp [range_subset_iff, funext_iff, mem_singleton]
#align set.range_subset_singleton Set.range_subset_singleton
theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s := by
rw [compl_eq_univ_diff, image_diff_preimage, image_univ]
#align set.image_compl_preimage Set.image_compl_preimage
/-- Any map `f : ι → β` factors through a map `rangeFactorization f : ι → range f`. -/
def rangeFactorization (f : ι → β) : ι → range f := fun i => ⟨f i, mem_range_self i⟩
#align set.range_factorization Set.rangeFactorization
theorem rangeFactorization_eq {f : ι → β} : Subtype.val ∘ rangeFactorization f = f :=
funext fun _ => rfl
#align set.range_factorization_eq Set.rangeFactorization_eq
@[simp]
theorem rangeFactorization_coe (f : ι → β) (a : ι) : (rangeFactorization f a : β) = f a :=
rfl
#align set.range_factorization_coe Set.rangeFactorization_coe
@[simp]
theorem coe_comp_rangeFactorization (f : ι → β) : (↑) ∘ rangeFactorization f = f := rfl
#align set.coe_comp_range_factorization Set.coe_comp_rangeFactorization
theorem surjective_onto_range : Surjective (rangeFactorization f) := fun ⟨_, ⟨i, rfl⟩⟩ => ⟨i, rfl⟩
#align set.surjective_onto_range Set.surjective_onto_range
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
constructor
|
rintro ⟨x, h1, h2⟩
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
constructor
|
Mathlib.Data.Set.Image.1093_0.IJFiTzmYGOCpPSd
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x
|
Mathlib_Data_Set_Image
|
case h.mp.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set α
x✝ : β
x : α
h1 : x ∈ s
h2 : f x = x✝
⊢ x✝ ∈ range fun x => f ↑x
case h.mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set α
x✝ : β
⊢ (x✝ ∈ range fun x => f ↑x) → x✝ ∈ f '' s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
exact hab
-- Porting note: `simp_rw` fails here
-- simp_rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.map, Subtype.coe_mk,
-- mem_set_of, exists_prop]
#align set.range_subtype_map Set.range_subtype_map
theorem image_swap_eq_preimage_swap : image (@Prod.swap α β) = preimage Prod.swap :=
image_eq_preimage_of_inverse Prod.swap_leftInverse Prod.swap_rightInverse
#align set.image_swap_eq_preimage_swap Set.image_swap_eq_preimage_swap
theorem preimage_singleton_nonempty {f : α → β} {y : β} : (f ⁻¹' {y}).Nonempty ↔ y ∈ range f :=
Iff.rfl
#align set.preimage_singleton_nonempty Set.preimage_singleton_nonempty
theorem preimage_singleton_eq_empty {f : α → β} {y : β} : f ⁻¹' {y} = ∅ ↔ y ∉ range f :=
not_nonempty_iff_eq_empty.symm.trans preimage_singleton_nonempty.not
#align set.preimage_singleton_eq_empty Set.preimage_singleton_eq_empty
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x := by
simp [range_subset_iff, funext_iff, mem_singleton]
#align set.range_subset_singleton Set.range_subset_singleton
theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s := by
rw [compl_eq_univ_diff, image_diff_preimage, image_univ]
#align set.image_compl_preimage Set.image_compl_preimage
/-- Any map `f : ι → β` factors through a map `rangeFactorization f : ι → range f`. -/
def rangeFactorization (f : ι → β) : ι → range f := fun i => ⟨f i, mem_range_self i⟩
#align set.range_factorization Set.rangeFactorization
theorem rangeFactorization_eq {f : ι → β} : Subtype.val ∘ rangeFactorization f = f :=
funext fun _ => rfl
#align set.range_factorization_eq Set.rangeFactorization_eq
@[simp]
theorem rangeFactorization_coe (f : ι → β) (a : ι) : (rangeFactorization f a : β) = f a :=
rfl
#align set.range_factorization_coe Set.rangeFactorization_coe
@[simp]
theorem coe_comp_rangeFactorization (f : ι → β) : (↑) ∘ rangeFactorization f = f := rfl
#align set.coe_comp_range_factorization Set.coe_comp_rangeFactorization
theorem surjective_onto_range : Surjective (rangeFactorization f) := fun ⟨_, ⟨i, rfl⟩⟩ => ⟨i, rfl⟩
#align set.surjective_onto_range Set.surjective_onto_range
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
constructor
rintro ⟨x, h1, h2⟩
|
exact ⟨⟨x, h1⟩, h2⟩
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
constructor
rintro ⟨x, h1, h2⟩
|
Mathlib.Data.Set.Image.1093_0.IJFiTzmYGOCpPSd
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x
|
Mathlib_Data_Set_Image
|
case h.mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set α
x✝ : β
⊢ (x✝ ∈ range fun x => f ↑x) → x✝ ∈ f '' s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
exact hab
-- Porting note: `simp_rw` fails here
-- simp_rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.map, Subtype.coe_mk,
-- mem_set_of, exists_prop]
#align set.range_subtype_map Set.range_subtype_map
theorem image_swap_eq_preimage_swap : image (@Prod.swap α β) = preimage Prod.swap :=
image_eq_preimage_of_inverse Prod.swap_leftInverse Prod.swap_rightInverse
#align set.image_swap_eq_preimage_swap Set.image_swap_eq_preimage_swap
theorem preimage_singleton_nonempty {f : α → β} {y : β} : (f ⁻¹' {y}).Nonempty ↔ y ∈ range f :=
Iff.rfl
#align set.preimage_singleton_nonempty Set.preimage_singleton_nonempty
theorem preimage_singleton_eq_empty {f : α → β} {y : β} : f ⁻¹' {y} = ∅ ↔ y ∉ range f :=
not_nonempty_iff_eq_empty.symm.trans preimage_singleton_nonempty.not
#align set.preimage_singleton_eq_empty Set.preimage_singleton_eq_empty
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x := by
simp [range_subset_iff, funext_iff, mem_singleton]
#align set.range_subset_singleton Set.range_subset_singleton
theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s := by
rw [compl_eq_univ_diff, image_diff_preimage, image_univ]
#align set.image_compl_preimage Set.image_compl_preimage
/-- Any map `f : ι → β` factors through a map `rangeFactorization f : ι → range f`. -/
def rangeFactorization (f : ι → β) : ι → range f := fun i => ⟨f i, mem_range_self i⟩
#align set.range_factorization Set.rangeFactorization
theorem rangeFactorization_eq {f : ι → β} : Subtype.val ∘ rangeFactorization f = f :=
funext fun _ => rfl
#align set.range_factorization_eq Set.rangeFactorization_eq
@[simp]
theorem rangeFactorization_coe (f : ι → β) (a : ι) : (rangeFactorization f a : β) = f a :=
rfl
#align set.range_factorization_coe Set.rangeFactorization_coe
@[simp]
theorem coe_comp_rangeFactorization (f : ι → β) : (↑) ∘ rangeFactorization f = f := rfl
#align set.coe_comp_range_factorization Set.coe_comp_rangeFactorization
theorem surjective_onto_range : Surjective (rangeFactorization f) := fun ⟨_, ⟨i, rfl⟩⟩ => ⟨i, rfl⟩
#align set.surjective_onto_range Set.surjective_onto_range
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
constructor
rintro ⟨x, h1, h2⟩
exact ⟨⟨x, h1⟩, h2⟩
|
rintro ⟨⟨x, h1⟩, h2⟩
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
constructor
rintro ⟨x, h1, h2⟩
exact ⟨⟨x, h1⟩, h2⟩
|
Mathlib.Data.Set.Image.1093_0.IJFiTzmYGOCpPSd
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x
|
Mathlib_Data_Set_Image
|
case h.mpr.intro.mk
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set α
x✝ : β
x : α
h1 : x ∈ s
h2 : (fun x => f ↑x) { val := x, property := h1 } = x✝
⊢ x✝ ∈ f '' s
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
exact hab
-- Porting note: `simp_rw` fails here
-- simp_rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.map, Subtype.coe_mk,
-- mem_set_of, exists_prop]
#align set.range_subtype_map Set.range_subtype_map
theorem image_swap_eq_preimage_swap : image (@Prod.swap α β) = preimage Prod.swap :=
image_eq_preimage_of_inverse Prod.swap_leftInverse Prod.swap_rightInverse
#align set.image_swap_eq_preimage_swap Set.image_swap_eq_preimage_swap
theorem preimage_singleton_nonempty {f : α → β} {y : β} : (f ⁻¹' {y}).Nonempty ↔ y ∈ range f :=
Iff.rfl
#align set.preimage_singleton_nonempty Set.preimage_singleton_nonempty
theorem preimage_singleton_eq_empty {f : α → β} {y : β} : f ⁻¹' {y} = ∅ ↔ y ∉ range f :=
not_nonempty_iff_eq_empty.symm.trans preimage_singleton_nonempty.not
#align set.preimage_singleton_eq_empty Set.preimage_singleton_eq_empty
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x := by
simp [range_subset_iff, funext_iff, mem_singleton]
#align set.range_subset_singleton Set.range_subset_singleton
theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s := by
rw [compl_eq_univ_diff, image_diff_preimage, image_univ]
#align set.image_compl_preimage Set.image_compl_preimage
/-- Any map `f : ι → β` factors through a map `rangeFactorization f : ι → range f`. -/
def rangeFactorization (f : ι → β) : ι → range f := fun i => ⟨f i, mem_range_self i⟩
#align set.range_factorization Set.rangeFactorization
theorem rangeFactorization_eq {f : ι → β} : Subtype.val ∘ rangeFactorization f = f :=
funext fun _ => rfl
#align set.range_factorization_eq Set.rangeFactorization_eq
@[simp]
theorem rangeFactorization_coe (f : ι → β) (a : ι) : (rangeFactorization f a : β) = f a :=
rfl
#align set.range_factorization_coe Set.rangeFactorization_coe
@[simp]
theorem coe_comp_rangeFactorization (f : ι → β) : (↑) ∘ rangeFactorization f = f := rfl
#align set.coe_comp_range_factorization Set.coe_comp_rangeFactorization
theorem surjective_onto_range : Surjective (rangeFactorization f) := fun ⟨_, ⟨i, rfl⟩⟩ => ⟨i, rfl⟩
#align set.surjective_onto_range Set.surjective_onto_range
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
constructor
rintro ⟨x, h1, h2⟩
exact ⟨⟨x, h1⟩, h2⟩
rintro ⟨⟨x, h1⟩, h2⟩
|
exact ⟨x, h1, h2⟩
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
constructor
rintro ⟨x, h1, h2⟩
exact ⟨⟨x, h1⟩, h2⟩
rintro ⟨⟨x, h1⟩, h2⟩
|
Mathlib.Data.Set.Image.1093_0.IJFiTzmYGOCpPSd
|
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x
|
Mathlib_Data_Set_Image
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : Prop
inst✝ : Decidable p
f g : α → β
⊢ range (if p then f else g) ⊆ range f ∪ range g
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
exact hab
-- Porting note: `simp_rw` fails here
-- simp_rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.map, Subtype.coe_mk,
-- mem_set_of, exists_prop]
#align set.range_subtype_map Set.range_subtype_map
theorem image_swap_eq_preimage_swap : image (@Prod.swap α β) = preimage Prod.swap :=
image_eq_preimage_of_inverse Prod.swap_leftInverse Prod.swap_rightInverse
#align set.image_swap_eq_preimage_swap Set.image_swap_eq_preimage_swap
theorem preimage_singleton_nonempty {f : α → β} {y : β} : (f ⁻¹' {y}).Nonempty ↔ y ∈ range f :=
Iff.rfl
#align set.preimage_singleton_nonempty Set.preimage_singleton_nonempty
theorem preimage_singleton_eq_empty {f : α → β} {y : β} : f ⁻¹' {y} = ∅ ↔ y ∉ range f :=
not_nonempty_iff_eq_empty.symm.trans preimage_singleton_nonempty.not
#align set.preimage_singleton_eq_empty Set.preimage_singleton_eq_empty
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x := by
simp [range_subset_iff, funext_iff, mem_singleton]
#align set.range_subset_singleton Set.range_subset_singleton
theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s := by
rw [compl_eq_univ_diff, image_diff_preimage, image_univ]
#align set.image_compl_preimage Set.image_compl_preimage
/-- Any map `f : ι → β` factors through a map `rangeFactorization f : ι → range f`. -/
def rangeFactorization (f : ι → β) : ι → range f := fun i => ⟨f i, mem_range_self i⟩
#align set.range_factorization Set.rangeFactorization
theorem rangeFactorization_eq {f : ι → β} : Subtype.val ∘ rangeFactorization f = f :=
funext fun _ => rfl
#align set.range_factorization_eq Set.rangeFactorization_eq
@[simp]
theorem rangeFactorization_coe (f : ι → β) (a : ι) : (rangeFactorization f a : β) = f a :=
rfl
#align set.range_factorization_coe Set.rangeFactorization_coe
@[simp]
theorem coe_comp_rangeFactorization (f : ι → β) : (↑) ∘ rangeFactorization f = f := rfl
#align set.coe_comp_range_factorization Set.coe_comp_rangeFactorization
theorem surjective_onto_range : Surjective (rangeFactorization f) := fun ⟨_, ⟨i, rfl⟩⟩ => ⟨i, rfl⟩
#align set.surjective_onto_range Set.surjective_onto_range
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
constructor
rintro ⟨x, h1, h2⟩
exact ⟨⟨x, h1⟩, h2⟩
rintro ⟨⟨x, h1⟩, h2⟩
exact ⟨x, h1, h2⟩
#align set.image_eq_range Set.image_eq_range
theorem _root_.Sum.range_eq (f : Sum α β → γ) :
range f = range (f ∘ Sum.inl) ∪ range (f ∘ Sum.inr) :=
ext fun _ => Sum.exists
#align sum.range_eq Sum.range_eq
@[simp]
theorem Sum.elim_range (f : α → γ) (g : β → γ) : range (Sum.elim f g) = range f ∪ range g :=
Sum.range_eq _
#align set.sum.elim_range Set.Sum.elim_range
theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g := by
|
by_cases h : p
|
theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g := by
|
Mathlib.Data.Set.Image.1112_0.IJFiTzmYGOCpPSd
|
theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g
|
Mathlib_Data_Set_Image
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : Prop
inst✝ : Decidable p
f g : α → β
h : p
⊢ range (if p then f else g) ⊆ range f ∪ range g
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Images and preimages of sets
## Main definitions
* `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β.
* `range f : Set β` : the image of `univ` under `f`.
Also works for `{p : Prop} (f : p → α)` (unlike `image`)
## Notation
* `f ⁻¹' t` for `Set.preimage f t`
* `f '' s` for `Set.image f s`
## Tags
set, sets, image, preimage, pre-image, range
-/
set_option autoImplicit true
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
/-! ### Inverse image -/
/-- The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`. -/
def preimage {α : Type u} {β : Type v} (f : α → β) (s : Set β) : Set α :=
{ x | f x ∈ s }
#align set.preimage Set.preimage
/-- `f ⁻¹' t` denotes the preimage of `t : Set β` under the function `f : α → β`. -/
infixl:80 " ⁻¹' " => preimage
section Preimage
variable {f : α → β} {g : β → γ}
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ :=
rfl
#align set.preimage_empty Set.preimage_empty
@[simp, mfld_simps]
theorem mem_preimage {s : Set β} {a : α} : a ∈ f ⁻¹' s ↔ f a ∈ s :=
Iff.rfl
#align set.mem_preimage Set.mem_preimage
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by
congr with x
simp [h]
#align set.preimage_congr Set.preimage_congr
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx
#align set.preimage_mono Set.preimage_mono
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ :=
rfl
#align set.preimage_univ Set.preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ :=
subset_univ _
#align set.subset_preimage_univ Set.subset_preimage_univ
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t :=
rfl
#align set.preimage_inter Set.preimage_inter
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t :=
rfl
#align set.preimage_union Set.preimage_union
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ :=
rfl
#align set.preimage_compl Set.preimage_compl
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t :=
rfl
#align set.preimage_diff Set.preimage_diff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=
rfl
#align set.preimage_symm_diff Set.preimage_symmDiff
@[simp]
theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) :
f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) :=
rfl
#align set.preimage_ite Set.preimage_ite
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } :=
rfl
#align set.preimage_set_of_eq Set.preimage_setOf_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id :=
rfl
#align set.preimage_id_eq Set.preimage_id_eq
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s :=
rfl
#align set.preimage_id Set.preimage_id
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s :=
rfl
#align set.preimage_id' Set.preimage_id'
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ :=
eq_univ_of_forall fun _ => h
#align set.preimage_const_of_mem Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ :=
eq_empty_of_subset_empty fun _ hx => h hx
#align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
#align set.preimage_const Set.preimage_const
/-- If preimage of each singleton under `f : α → β` is either empty or the whole type,
then `f` is a constant. -/
lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β}
(hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by
rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ | hf'
· exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩
· have : ∀ x b, f x ≠ b := fun x b ↦
eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x
exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) :=
rfl
#align set.preimage_comp Set.preimage_comp
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g :=
rfl
#align set.preimage_comp_eq Set.preimage_comp_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
#align set.preimage_iterate_eq Set.preimage_iterate_eq
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s :=
preimage_comp.symm
#align set.preimage_preimage Set.preimage_preimage
theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} :
s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t :=
⟨fun s_eq x h => by
rw [s_eq]
simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩
#align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
s.Nonempty :=
let ⟨x, hx⟩ := hf
⟨f x, hx⟩
#align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} := by ext; simp
#align set.preimage_singleton_true Set.preimage_singleton_true
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} := by ext; simp
#align set.preimage_singleton_false Set.preimage_singleton_false
theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
#align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl
end Preimage
/-! ### Image of a set under a function -/
section Image
variable {f : α → β} {s t : Set α}
-- Porting note: `Set.image` is already defined in `Init.Set`
#align set.image Set.image
/-- `f '' s` denotes the image of `s : Set α` under the function `f : α → β`. -/
infixl:80 " '' " => image
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y :=
bex_def.symm
#align set.mem_image_iff_bex Set.mem_image_iff_bex
@[simp]
theorem mem_image (f : α → β) (s : Set α) (y : β) : y ∈ f '' s ↔ ∃ x ∈ s, f x = y :=
Iff.rfl
#align set.mem_image Set.mem_image
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s :=
rfl
#align set.image_eta Set.image_eta
@[mfld_simps]
theorem mem_image_of_mem (f : α → β) {x : α} {a : Set α} (h : x ∈ a) : f x ∈ f '' a :=
⟨_, h, rfl⟩
#align set.mem_image_of_mem Set.mem_image_of_mem
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
f a ∈ f '' s ↔ a ∈ s :=
⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩
#align function.injective.mem_set_image Function.Injective.mem_set_image
theorem ball_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ x ∈ s, p (f x) := by simp
#align set.ball_image_iff Set.ball_image_iff
theorem ball_image_of_ball {f : α → β} {s : Set α} {p : β → Prop} (h : ∀ x ∈ s, p (f x)) :
∀ y ∈ f '' s, p y :=
ball_image_iff.2 h
#align set.ball_image_of_ball Set.ball_image_of_ball
theorem bex_image_iff {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp
#align set.bex_image_iff Set.bex_image_iff
theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) :
∀ {y : β}, y ∈ f '' s → C y
| _, ⟨a, a_in, rfl⟩ => h a a_in
#align set.mem_image_elim Set.mem_image_elim
theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s)
(h : ∀ x : α, x ∈ s → C (f x)) : C y :=
mem_image_elim h h_y
#align set.mem_image_elim_on Set.mem_image_elim_on
-- Porting note: used to be `safe`
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by
ext x
rw [mem_image, mem_image]
exact {
mp := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩,
mpr := by
rintro ⟨a, ha1, ha2⟩
exact ⟨a, ⟨ha1, (h a ha1) ▸ ha2⟩⟩
}
-- safe [ext_iff, iff_def]
#align set.image_congr Set.image_congr
/-- A common special case of `image_congr` -/
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s :=
image_congr fun x _ => h x
#align set.image_congr' Set.image_congr'
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by
rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha)
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) :=
Subset.antisymm (ball_image_of_ball fun _ ha => mem_image_of_mem _ <| mem_image_of_mem _ ha)
(ball_image_of_ball <| ball_image_of_ball fun _ ha => mem_image_of_mem _ ha)
#align set.image_comp Set.image_comp
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp
/-- A variant of `image_comp`, useful for rewriting -/
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s :=
(image_comp g f s).symm
#align set.image_image Set.image_image
theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by
simp_rw [image_image, h_comm]
#align set.image_comm Set.image_comm
theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β}
(h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
image_comm h
#align function.semiconj.set_image Function.Semiconj.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
Function.Commute (image f) (image g) :=
Function.Semiconj.set_image h
#align function.commute.set_image Function.Commute.set_image
/-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in
terms of `≤`. -/
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by
simp only [subset_def, mem_image]
exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩
#align set.image_subset Set.image_subset
/-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/
lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _
#align set.monotone_image Set.monotone_image
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t :=
ext fun x =>
⟨by rintro ⟨a, h | h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by
rintro (⟨a, h, rfl⟩ | ⟨a, h, rfl⟩) <;> refine' ⟨_, _, rfl⟩ <;> [left; right] <;> exact h⟩
#align set.image_union Set.image_union
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ := by
ext
simp
#align set.image_empty Set.image_empty
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
subset_inter (image_subset _ <| inter_subset_left _ _) (image_subset _ <| inter_subset_right _ _)
#align set.image_inter_subset Set.image_inter_subset
theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) :
f '' (s ∩ t) = f '' s ∩ f '' t :=
(image_inter_subset _ _ _).antisymm
fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦
have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp [*])
⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩
#align set.image_inter_on Set.image_inter_on
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t :=
image_inter_on fun _ _ _ _ h => H h
#align set.image_inter Set.image_inter
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
#align set.image_univ_of_surjective Set.image_univ_of_surjective
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by
ext
simp [image, eq_comm]
#align set.image_singleton Set.image_singleton
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} :=
ext fun _ =>
⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h =>
(eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩
#align set.nonempty.image_const Set.Nonempty.image_const
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by
simp only [eq_empty_iff_forall_not_mem]
exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩
#align set.image_eq_empty Set.image_eq_empty
-- Porting note: `compl` is already defined in `Init.Set`
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
HasCompl.compl ⁻¹' S = HasCompl.compl '' S :=
Set.ext fun x =>
⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h =>
Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩
#align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by
simp [← preimage_compl_eq_image_compl]
#align set.mem_compl_image Set.mem_compl_image
@[simp]
theorem image_id_eq : image (id : α → α) = id := by ext; simp
/-- A variant of `image_id` -/
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s := by
ext
simp
#align set.image_id' Set.image_id'
theorem image_id (s : Set α) : id '' s = s := by simp
#align set.image_id Set.image_id
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] := by
induction' n with n ih; · simp
rw [iterate_succ', iterate_succ',← ih, image_comp_eq]
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
#align set.compl_compl_image Set.compl_compl_image
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
ext
simp [and_or_left, exists_or, eq_comm, or_comm, and_comm]
#align set.image_insert_eq Set.image_insert_eq
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} := by
simp only [image_insert_eq, image_singleton]
#align set.image_pair Set.image_pair
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
f '' s ⊆ g ⁻¹' s := fun _ ⟨a, h, e⟩ => e ▸ ((I a).symm ▸ h : g (f a) ∈ s)
#align set.image_subset_preimage_of_inverse Set.image_subset_preimage_of_inverse
theorem preimage_subset_image_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set β) :
f ⁻¹' s ⊆ g '' s := fun b h => ⟨f b, h, I b⟩
#align set.preimage_subset_image_of_inverse Set.preimage_subset_image_of_inverse
theorem image_eq_preimage_of_inverse {f : α → β} {g : β → α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : image f = preimage g :=
funext fun s =>
Subset.antisymm (image_subset_preimage_of_inverse h₁ s) (preimage_subset_image_of_inverse h₂ s)
#align set.image_eq_preimage_of_inverse Set.image_eq_preimage_of_inverse
theorem mem_image_iff_of_inverse {f : α → β} {g : β → α} {b : β} {s : Set α} (h₁ : LeftInverse g f)
(h₂ : RightInverse g f) : b ∈ f '' s ↔ g b ∈ s := by
rw [image_eq_preimage_of_inverse h₁ h₂]; rfl
#align set.mem_image_iff_of_inverse Set.mem_image_iff_of_inverse
theorem image_compl_subset {f : α → β} {s : Set α} (H : Injective f) : f '' sᶜ ⊆ (f '' s)ᶜ :=
Disjoint.subset_compl_left <| by simp [disjoint_iff_inf_le, ← image_inter H]
#align set.image_compl_subset Set.image_compl_subset
theorem subset_image_compl {f : α → β} {s : Set α} (H : Surjective f) : (f '' s)ᶜ ⊆ f '' sᶜ :=
compl_subset_iff_union.2 <| by
rw [← image_union]
simp [image_univ_of_surjective H]
#align set.subset_image_compl Set.subset_image_compl
theorem image_compl_eq {f : α → β} {s : Set α} (H : Bijective f) : f '' sᶜ = (f '' s)ᶜ :=
Subset.antisymm (image_compl_subset H.1) (subset_image_compl H.2)
#align set.image_compl_eq Set.image_compl_eq
theorem subset_image_diff (f : α → β) (s t : Set α) : f '' s \ f '' t ⊆ f '' (s \ t) := by
rw [diff_subset_iff, ← image_union, union_diff_self]
exact image_subset f (subset_union_right t s)
#align set.subset_image_diff Set.subset_image_diff
theorem subset_image_symmDiff : (f '' s) ∆ (f '' t) ⊆ f '' s ∆ t :=
(union_subset_union (subset_image_diff _ _ _) <| subset_image_diff _ _ _).trans
(superset_of_eq (image_union _ _ _))
#align set.subset_image_symm_diff Set.subset_image_symmDiff
theorem image_diff {f : α → β} (hf : Injective f) (s t : Set α) : f '' (s \ t) = f '' s \ f '' t :=
Subset.antisymm
(Subset.trans (image_inter_subset _ _ _) <| inter_subset_inter_right _ <| image_compl_subset hf)
(subset_image_diff f s t)
#align set.image_diff Set.image_diff
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
simp_rw [Set.symmDiff_def, image_union, image_diff hf]
#align set.image_symm_diff Set.image_symmDiff
theorem Nonempty.image (f : α → β) {s : Set α} : s.Nonempty → (f '' s).Nonempty
| ⟨x, hx⟩ => ⟨f x, mem_image_of_mem f hx⟩
#align set.nonempty.image Set.Nonempty.image
theorem Nonempty.of_image {f : α → β} {s : Set α} : (f '' s).Nonempty → s.Nonempty
| ⟨_, x, hx, _⟩ => ⟨x, hx⟩
#align set.nonempty.of_image Set.Nonempty.of_image
@[simp]
theorem nonempty_image_iff {f : α → β} {s : Set α} : (f '' s).Nonempty ↔ s.Nonempty :=
⟨Nonempty.of_image, fun h => h.image f⟩
#align set.nonempty_image_iff Set.nonempty_image_iff
theorem Nonempty.preimage {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : Surjective f) :
(f ⁻¹' s).Nonempty :=
let ⟨y, hy⟩ := hs
let ⟨x, hx⟩ := hf y
⟨x, mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage Set.Nonempty.preimage
instance (f : α → β) (s : Set α) [Nonempty s] : Nonempty (f '' s) :=
(Set.Nonempty.image f nonempty_of_nonempty_subtype).to_subtype
/-- image and preimage are a Galois connection -/
@[simp]
theorem image_subset_iff {s : Set α} {t : Set β} {f : α → β} : f '' s ⊆ t ↔ s ⊆ f ⁻¹' t :=
ball_image_iff
#align set.image_subset_iff Set.image_subset_iff
theorem image_preimage_subset (f : α → β) (s : Set β) : f '' (f ⁻¹' s) ⊆ s :=
image_subset_iff.2 Subset.rfl
#align set.image_preimage_subset Set.image_preimage_subset
theorem subset_preimage_image (f : α → β) (s : Set α) : s ⊆ f ⁻¹' (f '' s) := fun _ =>
mem_image_of_mem f
#align set.subset_preimage_image Set.subset_preimage_image
@[simp]
theorem preimage_image_eq {f : α → β} (s : Set α) (h : Injective f) : f ⁻¹' (f '' s) = s :=
Subset.antisymm (fun _ ⟨_, hy, e⟩ => h e ▸ hy) (subset_preimage_image f s)
#align set.preimage_image_eq Set.preimage_image_eq
@[simp]
theorem image_preimage_eq {f : α → β} (s : Set β) (h : Surjective f) : f '' (f ⁻¹' s) = s :=
Subset.antisymm (image_preimage_subset f s) fun x hx =>
let ⟨y, e⟩ := h x
⟨y, (e.symm ▸ hx : f y ∈ s), e⟩
#align set.image_preimage_eq Set.image_preimage_eq
@[simp]
theorem preimage_eq_preimage {f : β → α} (hf : Surjective f) : f ⁻¹' s = f ⁻¹' t ↔ s = t :=
Iff.intro
fun eq => by rw [← image_preimage_eq s hf, ← image_preimage_eq t hf, eq]
fun eq => eq ▸ rfl
#align set.preimage_eq_preimage Set.preimage_eq_preimage
theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
#align set.image_inter_preimage Set.image_inter_preimage
theorem image_preimage_inter (f : α → β) (s : Set α) (t : Set β) :
f '' (f ⁻¹' t ∩ s) = t ∩ f '' s := by simp only [inter_comm, image_inter_preimage]
#align set.image_preimage_inter Set.image_preimage_inter
@[simp]
theorem image_inter_nonempty_iff {f : α → β} {s : Set α} {t : Set β} :
(f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty := by
rw [← image_inter_preimage, nonempty_image_iff]
#align set.image_inter_nonempty_iff Set.image_inter_nonempty_iff
theorem image_diff_preimage {f : α → β} {s : Set α} {t : Set β} : f '' (s \ f ⁻¹' t) = f '' s \ t :=
by simp_rw [diff_eq, ← preimage_compl, image_inter_preimage]
#align set.image_diff_preimage Set.image_diff_preimage
theorem compl_image : image (compl : Set α → Set α) = preimage compl :=
image_eq_preimage_of_inverse compl_compl compl_compl
#align set.compl_image Set.compl_image
theorem compl_image_set_of {p : Set α → Prop} : compl '' { s | p s } = { s | p sᶜ } :=
congr_fun compl_image p
#align set.compl_image_set_of Set.compl_image_set_of
theorem inter_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t) := fun _ h => ⟨mem_image_of_mem _ h.left, h.right⟩
#align set.inter_preimage_subset Set.inter_preimage_subset
theorem union_preimage_subset (s : Set α) (t : Set β) (f : α → β) :
s ∪ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∪ t) := fun _ h =>
Or.elim h (fun l => Or.inl <| mem_image_of_mem _ l) fun r => Or.inr r
#align set.union_preimage_subset Set.union_preimage_subset
theorem subset_image_union (f : α → β) (s : Set α) (t : Set β) : f '' (s ∪ f ⁻¹' t) ⊆ f '' s ∪ t :=
image_subset_iff.2 (union_preimage_subset _ _ _)
#align set.subset_image_union Set.subset_image_union
theorem preimage_subset_iff {A : Set α} {B : Set β} {f : α → β} :
f ⁻¹' B ⊆ A ↔ ∀ a : α, f a ∈ B → a ∈ A :=
Iff.rfl
#align set.preimage_subset_iff Set.preimage_subset_iff
theorem image_eq_image {f : α → β} (hf : Injective f) : f '' s = f '' t ↔ s = t :=
Iff.symm <|
(Iff.intro fun eq => eq ▸ rfl) fun eq => by
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]
#align set.image_eq_image Set.image_eq_image
theorem image_subset_image_iff {f : α → β} (hf : Injective f) : f '' s ⊆ f '' t ↔ s ⊆ t := by
refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf]
exact preimage_mono h
#align set.image_subset_image_iff Set.image_subset_image_iff
theorem prod_quotient_preimage_eq_image [s : Setoid α] (g : Quotient s → β) {h : α → β}
(Hh : h = g ∘ Quotient.mk'') (r : Set (β × β)) :
{ x : Quotient s × Quotient s | (g x.1, g x.2) ∈ r } =
(fun a : α × α => (⟦a.1⟧, ⟦a.2⟧)) '' ((fun a : α × α => (h a.1, h a.2)) ⁻¹' r) :=
Hh.symm ▸
Set.ext fun ⟨a₁, a₂⟩ =>
⟨Quot.induction_on₂ a₁ a₂ fun a₁ a₂ h => ⟨(a₁, a₂), h, rfl⟩, fun ⟨⟨b₁, b₂⟩, h₁, h₂⟩ =>
show (g a₁, g a₂) ∈ r from
have h₃ : ⟦b₁⟧ = a₁ ∧ ⟦b₂⟧ = a₂ := Prod.ext_iff.1 h₂
h₃.1 ▸ h₃.2 ▸ h₁⟩
#align set.prod_quotient_preimage_eq_image Set.prod_quotient_preimage_eq_image
theorem exists_image_iff (f : α → β) (x : Set α) (P : β → Prop) :
(∃ a : f '' x, P a) ↔ ∃ a : x, P (f a) :=
⟨fun ⟨a, h⟩ => ⟨⟨_, a.prop.choose_spec.1⟩, a.prop.choose_spec.2.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨⟨_, _, a.prop, rfl⟩, h⟩⟩
#align set.exists_image_iff Set.exists_image_iff
/-- Restriction of `f` to `s` factors through `s.imageFactorization f : s → f '' s`. -/
def imageFactorization (f : α → β) (s : Set α) : s → f '' s := fun p =>
⟨f p.1, mem_image_of_mem f p.2⟩
#align set.image_factorization Set.imageFactorization
theorem imageFactorization_eq {f : α → β} {s : Set α} :
Subtype.val ∘ imageFactorization f s = f ∘ Subtype.val :=
funext fun _ => rfl
#align set.image_factorization_eq Set.imageFactorization_eq
theorem surjective_onto_image {f : α → β} {s : Set α} : Surjective (imageFactorization f s) :=
fun ⟨_, ⟨a, ha, rfl⟩⟩ => ⟨⟨a, ha⟩, rfl⟩
#align set.surjective_onto_image Set.surjective_onto_image
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem image_perm {s : Set α} {σ : Equiv.Perm α} (hs : { a : α | σ a ≠ a } ⊆ s) : σ '' s = s := by
ext i
obtain hi | hi := eq_or_ne (σ i) i
· refine' ⟨_, fun h => ⟨i, h, hi⟩⟩
rintro ⟨j, hj, h⟩
rwa [σ.injective (hi.trans h.symm)]
· refine' iff_of_true ⟨σ.symm i, hs fun h => hi _, σ.apply_symm_apply _⟩ (hs hi)
convert congr_arg σ h <;> exact (σ.apply_symm_apply _).symm
#align set.image_perm Set.image_perm
end Image
/-! ### Lemmas about the powerset and image. -/
/-- The powerset of `{a} ∪ s` is `𝒫 s` together with `{a} ∪ t` for each `t ∈ 𝒫 s`. -/
theorem powerset_insert (s : Set α) (a : α) : 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s := by
ext t
simp_rw [mem_union, mem_image, mem_powerset_iff]
constructor
· intro h
by_cases hs : a ∈ t
· right
refine' ⟨t \ {a}, _, _⟩
· rw [diff_singleton_subset_iff]
assumption
· rw [insert_diff_singleton, insert_eq_of_mem hs]
· left
exact (subset_insert_iff_of_not_mem hs).mp h
· rintro (h | ⟨s', h₁, rfl⟩)
· exact subset_trans h (subset_insert a s)
· exact insert_subset_insert h₁
#align set.powerset_insert Set.powerset_insert
/-! ### Lemmas about range of a function. -/
section Range
variable {f : ι → α} {s t : Set α}
/-- Range of a function.
This function is more flexible than `f '' univ`, as the image requires that the domain is in Type
and not an arbitrary Sort. -/
def range (f : ι → α) : Set α :=
{ x | ∃ y, f y = x }
#align set.range Set.range
@[simp]
theorem mem_range {x : α} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
#align set.mem_range Set.mem_range
-- Porting note
-- @[simp] `simp` can prove this
@[mfld_simps]
theorem mem_range_self (i : ι) : f i ∈ range f :=
⟨i, rfl⟩
#align set.mem_range_self Set.mem_range_self
theorem forall_range_iff {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by simp
#align set.forall_range_iff Set.forall_range_iff
theorem forall_subtype_range_iff {p : range f → Prop} :
(∀ a : range f, p a) ↔ ∀ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun H i => H _, fun H ⟨y, i, hi⟩ => by
subst hi
apply H⟩
#align set.forall_subtype_range_iff Set.forall_subtype_range_iff
theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by simp
#align set.exists_range_iff Set.exists_range_iff
theorem exists_range_iff' {p : α → Prop} : (∃ a, a ∈ range f ∧ p a) ↔ ∃ i, p (f i) := by
simpa only [exists_prop] using exists_range_iff
#align set.exists_range_iff' Set.exists_range_iff'
theorem exists_subtype_range_iff {p : range f → Prop} :
(∃ a : range f, p a) ↔ ∃ i, p ⟨f i, mem_range_self _⟩ :=
⟨fun ⟨⟨a, i, hi⟩, ha⟩ => by
subst a
exact ⟨i, ha⟩,
fun ⟨i, hi⟩ => ⟨_, hi⟩⟩
#align set.exists_subtype_range_iff Set.exists_subtype_range_iff
theorem range_iff_surjective : range f = univ ↔ Surjective f :=
eq_univ_iff_forall
#align set.range_iff_surjective Set.range_iff_surjective
-- Porting note: Lean4 unfolds `Surjective` here, ruining dot notation
alias ⟨_, _root_.Function.Surjective.range_eq⟩ := range_iff_surjective
#align function.surjective.range_eq Function.Surjective.range_eq
@[simp]
theorem image_univ {f : α → β} : f '' univ = range f := by
ext
simp [image, range]
#align set.image_univ Set.image_univ
theorem image_subset_range (f : α → β) (s) : f '' s ⊆ range f := by
rw [← image_univ]; exact image_subset _ (subset_univ _)
#align set.image_subset_range Set.image_subset_range
theorem mem_range_of_mem_image (f : α → β) (s) {x : β} (h : x ∈ f '' s) : x ∈ range f :=
image_subset_range f s h
#align set.mem_range_of_mem_image Set.mem_range_of_mem_image
theorem _root_.Nat.mem_range_succ (i : ℕ) : i ∈ range Nat.succ ↔ 0 < i :=
⟨by
rintro ⟨n, rfl⟩
exact Nat.succ_pos n, fun h => ⟨_, Nat.succ_pred_eq_of_pos h⟩⟩
#align nat.mem_range_succ Nat.mem_range_succ
theorem Nonempty.preimage' {s : Set β} (hs : s.Nonempty) {f : α → β} (hf : s ⊆ range f) :
(f ⁻¹' s).Nonempty :=
let ⟨_, hy⟩ := hs
let ⟨x, hx⟩ := hf hy
⟨x, Set.mem_preimage.2 <| hx.symm ▸ hy⟩
#align set.nonempty.preimage' Set.Nonempty.preimage'
theorem range_comp (g : α → β) (f : ι → α) : range (g ∘ f) = g '' range f :=
Subset.antisymm (forall_range_iff.mpr fun _ => mem_image_of_mem g (mem_range_self _))
(ball_image_iff.mpr <| forall_range_iff.mpr mem_range_self)
#align set.range_comp Set.range_comp
theorem range_subset_iff : range f ⊆ s ↔ ∀ y, f y ∈ s :=
forall_range_iff
#align set.range_subset_iff Set.range_subset_iff
theorem range_subset_range_iff_exists_comp {f : α → γ} {g : β → γ} :
range f ⊆ range g ↔ ∃ h : α → β, f = g ∘ h := by
simp only [range_subset_iff, mem_range, Classical.skolem, Function.funext_iff, (· ∘ ·), eq_comm]
theorem range_eq_iff (f : α → β) (s : Set β) :
range f = s ↔ (∀ a, f a ∈ s) ∧ ∀ b ∈ s, ∃ a, f a = b := by
rw [← range_subset_iff]
exact le_antisymm_iff
#align set.range_eq_iff Set.range_eq_iff
theorem range_comp_subset_range (f : α → β) (g : β → γ) : range (g ∘ f) ⊆ range g := by
rw [range_comp]; apply image_subset_range
#align set.range_comp_subset_range Set.range_comp_subset_range
theorem range_nonempty_iff_nonempty : (range f).Nonempty ↔ Nonempty ι :=
⟨fun ⟨_, x, _⟩ => ⟨x⟩, fun ⟨x⟩ => ⟨f x, mem_range_self x⟩⟩
#align set.range_nonempty_iff_nonempty Set.range_nonempty_iff_nonempty
theorem range_nonempty [h : Nonempty ι] (f : ι → α) : (range f).Nonempty :=
range_nonempty_iff_nonempty.2 h
#align set.range_nonempty Set.range_nonempty
@[simp]
theorem range_eq_empty_iff {f : ι → α} : range f = ∅ ↔ IsEmpty ι := by
rw [← not_nonempty_iff, ← range_nonempty_iff_nonempty, not_nonempty_iff_eq_empty]
#align set.range_eq_empty_iff Set.range_eq_empty_iff
theorem range_eq_empty [IsEmpty ι] (f : ι → α) : range f = ∅ :=
range_eq_empty_iff.2 ‹_›
#align set.range_eq_empty Set.range_eq_empty
instance instNonemptyRange [Nonempty ι] (f : ι → α) : Nonempty (range f) :=
(range_nonempty f).to_subtype
@[simp]
theorem image_union_image_compl_eq_range (f : α → β) : f '' s ∪ f '' sᶜ = range f := by
rw [← image_union, ← image_univ, ← union_compl_self]
#align set.image_union_image_compl_eq_range Set.image_union_image_compl_eq_range
theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
#align set.insert_image_compl_eq_range Set.insert_image_compl_eq_range
theorem image_preimage_eq_inter_range {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = t ∩ range f :=
ext fun x =>
⟨fun ⟨x, hx, HEq⟩ => HEq ▸ ⟨hx, mem_range_self _⟩, fun ⟨hx, ⟨y, h_eq⟩⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
#align set.image_preimage_eq_inter_range Set.image_preimage_eq_inter_range
theorem image_preimage_eq_of_subset {f : α → β} {s : Set β} (hs : s ⊆ range f) :
f '' (f ⁻¹' s) = s := by rw [image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
#align set.image_preimage_eq_of_subset Set.image_preimage_eq_of_subset
theorem image_preimage_eq_iff {f : α → β} {s : Set β} : f '' (f ⁻¹' s) = s ↔ s ⊆ range f :=
⟨by
intro h
rw [← h]
apply image_subset_range,
image_preimage_eq_of_subset⟩
#align set.image_preimage_eq_iff Set.image_preimage_eq_iff
theorem subset_range_iff_exists_image_eq {f : α → β} {s : Set β} : s ⊆ range f ↔ ∃ t, f '' t = s :=
⟨fun h => ⟨_, image_preimage_eq_iff.2 h⟩, fun ⟨_, ht⟩ => ht ▸ image_subset_range _ _⟩
#align set.subset_range_iff_exists_image_eq Set.subset_range_iff_exists_image_eq
theorem range_image (f : α → β) : range (image f) = 𝒫 range f :=
ext fun _ => subset_range_iff_exists_image_eq.symm
#align set.range_image Set.range_image
@[simp]
theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by
rw [← exists_range_iff, range_image]; rfl
#align set.exists_subset_range_and_iff Set.exists_subset_range_and_iff
theorem exists_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∃ (s : _) (_ : s ⊆ range f), p s) ↔ ∃ s, p (f '' s) := by simp
#align set.exists_subset_range_iff Set.exists_subset_range_iff
theorem forall_subset_range_iff {f : α → β} {p : Set β → Prop} :
(∀ s, s ⊆ range f → p s) ↔ ∀ s, p (f '' s) := by
rw [← forall_range_iff, range_image]; rfl
theorem preimage_subset_preimage_iff {s t : Set α} {f : β → α} (hs : s ⊆ range f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
constructor
· intro h x hx
rcases hs hx with ⟨y, rfl⟩
exact h hx
intro h x; apply h
#align set.preimage_subset_preimage_iff Set.preimage_subset_preimage_iff
theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t := by
constructor
· intro h
apply Subset.antisymm
· rw [← preimage_subset_preimage_iff hs, h]
· rw [← preimage_subset_preimage_iff ht, h]
rintro rfl; rfl
#align set.preimage_eq_preimage' Set.preimage_eq_preimage'
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_inter_range {f : α → β} {s : Set β} : f ⁻¹' (s ∩ range f) = f ⁻¹' s :=
Set.ext fun x => and_iff_left ⟨x, rfl⟩
#align set.preimage_inter_range Set.preimage_inter_range
-- Porting note:
-- @[simp] `simp` can prove this
theorem preimage_range_inter {f : α → β} {s : Set β} : f ⁻¹' (range f ∩ s) = f ⁻¹' s := by
rw [inter_comm, preimage_inter_range]
#align set.preimage_range_inter Set.preimage_range_inter
theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by
rw [image_preimage_eq_inter_range, preimage_inter_range]
#align set.preimage_image_preimage Set.preimage_image_preimage
@[simp, mfld_simps]
theorem range_id : range (@id α) = univ :=
range_iff_surjective.2 surjective_id
#align set.range_id Set.range_id
@[simp, mfld_simps]
theorem range_id' : (range fun x : α => x) = univ :=
range_id
#align set.range_id' Set.range_id'
@[simp]
theorem _root_.Prod.range_fst [Nonempty β] : range (Prod.fst : α × β → α) = univ :=
Prod.fst_surjective.range_eq
#align prod.range_fst Prod.range_fst
@[simp]
theorem _root_.Prod.range_snd [Nonempty α] : range (Prod.snd : α × β → β) = univ :=
Prod.snd_surjective.range_eq
#align prod.range_snd Prod.range_snd
@[simp]
theorem range_eval {α : ι → Sort _} [∀ i, Nonempty (α i)] (i : ι) :
range (eval i : (∀ i, α i) → α i) = univ :=
(surjective_eval i).range_eq
#align set.range_eval Set.range_eval
theorem range_inl : range (@Sum.inl α β) = {x | Sum.isLeft x} := by ext (_|_) <;> simp
#align set.range_inl Set.range_inl
theorem range_inr : range (@Sum.inr α β) = {x | Sum.isRight x} := by ext (_|_) <;> simp
#align set.range_inr Set.range_inr
theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
#align set.is_compl_range_inl_range_inr Set.isCompl_range_inl_range_inr
@[simp]
theorem range_inl_union_range_inr : range (Sum.inl : α → Sum α β) ∪ range Sum.inr = univ :=
isCompl_range_inl_range_inr.sup_eq_top
#align set.range_inl_union_range_inr Set.range_inl_union_range_inr
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → Sum α β) ∩ range Sum.inr = ∅ :=
isCompl_range_inl_range_inr.inf_eq_bot
#align set.range_inl_inter_range_inr Set.range_inl_inter_range_inr
@[simp]
theorem range_inr_union_range_inl : range (Sum.inr : β → Sum α β) ∪ range Sum.inl = univ :=
isCompl_range_inl_range_inr.symm.sup_eq_top
#align set.range_inr_union_range_inl Set.range_inr_union_range_inl
@[simp]
theorem range_inr_inter_range_inl : range (Sum.inr : β → Sum α β) ∩ range Sum.inl = ∅ :=
isCompl_range_inl_range_inr.symm.inf_eq_bot
#align set.range_inr_inter_range_inl Set.range_inr_inter_range_inl
@[simp]
theorem preimage_inl_image_inr (s : Set β) : Sum.inl ⁻¹' (@Sum.inr α β '' s) = ∅ := by
ext
simp
#align set.preimage_inl_image_inr Set.preimage_inl_image_inr
@[simp]
theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by
ext
simp
#align set.preimage_inr_image_inl Set.preimage_inr_image_inl
@[simp]
theorem preimage_inl_range_inr : Sum.inl ⁻¹' range (Sum.inr : β → Sum α β) = ∅ := by
rw [← image_univ, preimage_inl_image_inr]
#align set.preimage_inl_range_inr Set.preimage_inl_range_inr
@[simp]
theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by
rw [← image_univ, preimage_inr_image_inl]
#align set.preimage_inr_range_inl Set.preimage_inr_range_inl
@[simp]
theorem compl_range_inl : (range (Sum.inl : α → Sum α β))ᶜ = range (Sum.inr : β → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr
#align set.compl_range_inl Set.compl_range_inl
@[simp]
theorem compl_range_inr : (range (Sum.inr : β → Sum α β))ᶜ = range (Sum.inl : α → Sum α β) :=
IsCompl.compl_eq isCompl_range_inl_range_inr.symm
#align set.compl_range_inr Set.compl_range_inr
theorem image_preimage_inl_union_image_preimage_inr (s : Set (Sum α β)) :
Sum.inl '' (Sum.inl ⁻¹' s) ∪ Sum.inr '' (Sum.inr ⁻¹' s) = s := by
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, ← inter_distrib_left,
range_inl_union_range_inr, inter_univ]
#align set.image_preimage_inl_union_image_preimage_inr Set.image_preimage_inl_union_image_preimage_inr
@[simp]
theorem range_quot_mk (r : α → α → Prop) : range (Quot.mk r) = univ :=
(surjective_quot_mk r).range_eq
#align set.range_quot_mk Set.range_quot_mk
@[simp]
theorem range_quot_lift {r : ι → ι → Prop} (hf : ∀ x y, r x y → f x = f y) :
range (Quot.lift f hf) = range f :=
ext fun _ => (surjective_quot_mk _).exists
#align set.range_quot_lift Set.range_quot_lift
-- Porting note: the `Setoid α` instance is not being filled in
@[simp]
theorem range_quotient_mk [sa : Setoid α] : (range (α := Quotient sa) fun x : α => ⟦x⟧) = univ :=
range_quot_mk _
#align set.range_quotient_mk Set.range_quotient_mk
@[simp]
theorem range_quotient_lift [s : Setoid ι] (hf) :
range (Quotient.lift f hf : Quotient s → α) = range f :=
range_quot_lift _
#align set.range_quotient_lift Set.range_quotient_lift
@[simp]
theorem range_quotient_mk' {s : Setoid α} : range (Quotient.mk' : α → Quotient s) = univ :=
range_quot_mk _
#align set.range_quotient_mk' Set.range_quotient_mk'
@[simp] lemma Quotient.range_mk'' {sa : Setoid α} : range (Quotient.mk'' (s₁ := sa)) = univ :=
range_quotient_mk
@[simp]
theorem range_quotient_lift_on' {s : Setoid ι} (hf) :
(range fun x : Quotient s => Quotient.liftOn' x f hf) = range f :=
range_quot_lift _
#align set.range_quotient_lift_on' Set.range_quotient_lift_on'
instance canLift (c) (p) [CanLift α β c p] :
CanLift (Set α) (Set β) (c '' ·) fun s => ∀ x ∈ s, p x where
prf _ hs := subset_range_iff_exists_image_eq.mp fun x hx => CanLift.prf _ (hs x hx)
#align set.can_lift Set.canLift
theorem range_const_subset {c : α} : (range fun _ : ι => c) ⊆ {c} :=
range_subset_iff.2 fun _ => rfl
#align set.range_const_subset Set.range_const_subset
@[simp]
theorem range_const : ∀ [Nonempty ι] {c : α}, (range fun _ : ι => c) = {c}
| ⟨x⟩, _ =>
(Subset.antisymm range_const_subset) fun _ hy =>
(mem_singleton_iff.1 hy).symm ▸ mem_range_self x
#align set.range_const Set.range_const
theorem range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) :
range (Subtype.map f h) = (↑) ⁻¹' (f '' { x | p x }) := by
ext ⟨x, hx⟩
rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]
apply Iff.intro
· rintro ⟨a, b, hab⟩
rw [Subtype.map, Subtype.mk.injEq] at hab
use a
trivial
· rintro ⟨a, b, hab⟩
use a
use b
rw [Subtype.map, Subtype.mk.injEq]
exact hab
-- Porting note: `simp_rw` fails here
-- simp_rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.map, Subtype.coe_mk,
-- mem_set_of, exists_prop]
#align set.range_subtype_map Set.range_subtype_map
theorem image_swap_eq_preimage_swap : image (@Prod.swap α β) = preimage Prod.swap :=
image_eq_preimage_of_inverse Prod.swap_leftInverse Prod.swap_rightInverse
#align set.image_swap_eq_preimage_swap Set.image_swap_eq_preimage_swap
theorem preimage_singleton_nonempty {f : α → β} {y : β} : (f ⁻¹' {y}).Nonempty ↔ y ∈ range f :=
Iff.rfl
#align set.preimage_singleton_nonempty Set.preimage_singleton_nonempty
theorem preimage_singleton_eq_empty {f : α → β} {y : β} : f ⁻¹' {y} = ∅ ↔ y ∉ range f :=
not_nonempty_iff_eq_empty.symm.trans preimage_singleton_nonempty.not
#align set.preimage_singleton_eq_empty Set.preimage_singleton_eq_empty
theorem range_subset_singleton {f : ι → α} {x : α} : range f ⊆ {x} ↔ f = const ι x := by
simp [range_subset_iff, funext_iff, mem_singleton]
#align set.range_subset_singleton Set.range_subset_singleton
theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s := by
rw [compl_eq_univ_diff, image_diff_preimage, image_univ]
#align set.image_compl_preimage Set.image_compl_preimage
/-- Any map `f : ι → β` factors through a map `rangeFactorization f : ι → range f`. -/
def rangeFactorization (f : ι → β) : ι → range f := fun i => ⟨f i, mem_range_self i⟩
#align set.range_factorization Set.rangeFactorization
theorem rangeFactorization_eq {f : ι → β} : Subtype.val ∘ rangeFactorization f = f :=
funext fun _ => rfl
#align set.range_factorization_eq Set.rangeFactorization_eq
@[simp]
theorem rangeFactorization_coe (f : ι → β) (a : ι) : (rangeFactorization f a : β) = f a :=
rfl
#align set.range_factorization_coe Set.rangeFactorization_coe
@[simp]
theorem coe_comp_rangeFactorization (f : ι → β) : (↑) ∘ rangeFactorization f = f := rfl
#align set.coe_comp_range_factorization Set.coe_comp_rangeFactorization
theorem surjective_onto_range : Surjective (rangeFactorization f) := fun ⟨_, ⟨i, rfl⟩⟩ => ⟨i, rfl⟩
#align set.surjective_onto_range Set.surjective_onto_range
theorem image_eq_range (f : α → β) (s : Set α) : f '' s = range fun x : s => f x := by
ext
constructor
rintro ⟨x, h1, h2⟩
exact ⟨⟨x, h1⟩, h2⟩
rintro ⟨⟨x, h1⟩, h2⟩
exact ⟨x, h1, h2⟩
#align set.image_eq_range Set.image_eq_range
theorem _root_.Sum.range_eq (f : Sum α β → γ) :
range f = range (f ∘ Sum.inl) ∪ range (f ∘ Sum.inr) :=
ext fun _ => Sum.exists
#align sum.range_eq Sum.range_eq
@[simp]
theorem Sum.elim_range (f : α → γ) (g : β → γ) : range (Sum.elim f g) = range f ∪ range g :=
Sum.range_eq _
#align set.sum.elim_range Set.Sum.elim_range
theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g := by
by_cases h : p
·
|
rw [if_pos h]
|
theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g := by
by_cases h : p
·
|
Mathlib.Data.Set.Image.1112_0.IJFiTzmYGOCpPSd
|
theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g
|
Mathlib_Data_Set_Image
|
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