ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.pi_update_of_not_mem
theorem pi_update_of_not_mem [DecidableEq ι] (hi : i ∉ s) (f : ∀ j, α j) (a : α i)
    (t : ∀ j, α j → Set (β j)) : (s.pi fun j => t j (update f i a j)) = s.pi fun j => t j (f j) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {β : ι → Type u_3} {s : Set ι} {i : ι} [DecidableEq ι] (hi : i ∉ s) (f : (j : ι) → α j) (a : α i) (t : (j : ι) → α j → Set (β j))
Goal: (s.pi fun j => t j (Function.update f i a j)) = s.pi fun j => t j (f j)
-- Set.pi_update_of_mem
theorem pi_update_of_mem [DecidableEq ι] (hi : i ∈ s) (f : ∀ j, α j) (a : α i)
    (t : ∀ j, α j → Set (β j)) :
    (s.pi fun j => t j (update f i a j)) = { x | x i ∈ t i a } ∩ (s \ {i}).pi fun j => t j (f j) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {β : ι → Type u_3} {s : Set ι} {i : ι} [DecidableEq ι] (hi : i ∈ s) (f : (j : ι) → α j) (a : α i) (t : (j : ι) → α j → Set (β j))
Goal: (s.pi fun j => t j (Function.update f i a j)) = {x | x i ∈ t i a} ∩ (s \ {i}).pi fun j => t j (f j)
-- Set.univ_pi_update
theorem univ_pi_update [DecidableEq ι] {β : ι → Type*} (i : ι) (f : ∀ j, α j) (a : α i)
    (t : ∀ j, α j → Set (β j)) :
    (pi univ fun j => t j (update f i a j)) = { x | x i ∈ t i a } ∩ pi {i}ᶜ fun j => t j (f j) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] {β : ι → Type u_4} (i : ι) (f : (j : ι) → α j) (a : α i) (t : (j : ι) → α j → Set (β j))
Goal: (Set.univ.pi fun j => t j (Function.update f i a j)) = {x | x i ∈ t i a} ∩ {i}ᶜ.pi fun j => t j (f j)
-- Set.univ_pi_update_univ
theorem univ_pi_update_univ [DecidableEq ι] (i : ι) (s : Set (α i)) :
    pi univ (update (fun j : ι => (univ : Set (α j))) i s) = eval i ⁻¹' s 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] (i : ι) (s : Set (α i))
Goal: Set.univ.pi (Function.update (fun j => Set.univ) i s) = Function.eval i ⁻¹' s
-- Set.eval_image_pi_subset
theorem eval_image_pi_subset (hs : i ∈ s) : eval i '' s.pi t ⊆ t i 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} (hs : i ∈ s)
Goal: Function.eval i '' s.pi t ⊆ t i
-- Set.eval_image_univ_pi_subset
theorem eval_image_univ_pi_subset : eval i '' pi univ t ⊆ t i 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {t : (i : ι) → Set (α i)} {i : ι}
Goal: Function.eval i '' Set.univ.pi t ⊆ t i
-- Set.subset_eval_image_pi
theorem subset_eval_image_pi (ht : (s.pi t).Nonempty) (i : ι) : t i ⊆ eval i '' s.pi t 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} (ht : (s.pi t).Nonempty) (i : ι)
Goal: t i ⊆ Function.eval i '' s.pi t
-- Set.eval_image_pi
theorem eval_image_pi (hs : i ∈ s) (ht : (s.pi t).Nonempty) : eval i '' s.pi t = t i 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} (hs : i ∈ s) (ht : (s.pi t).Nonempty)
Goal: Function.eval i '' s.pi t = t i
-- Set.eval_image_pi_of_not_mem
lemma eval_image_pi_of_not_mem [Decidable (s.pi t).Nonempty] (hi : i ∉ s) :
    eval i '' s.pi t = if (s.pi t).Nonempty then univ else ∅ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} [Decidable (s.pi t).Nonempty] (hi : i ∉ s)
Goal: Function.eval i '' s.pi t = if (s.pi t).Nonempty then Set.univ else ∅
-- Set.eval_image_univ_pi
@[simp]
theorem eval_image_univ_pi (ht : (pi univ t).Nonempty) :
    (fun f : ∀ i, α i => f i) '' pi univ t = t i 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {t : (i : ι) → Set (α i)} {i : ι} (ht : (Set.univ.pi t).Nonempty)
Goal: (fun f => f i) '' Set.univ.pi t = t i
-- Set.pi_subset_pi_iff
theorem pi_subset_pi_iff : pi s t₁ ⊆ pi s t₂ ↔ (∀ i ∈ s, t₁ i ⊆ t₂ i) ∨ pi s t₁ = ∅ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t₁ t₂ : (i : ι) → Set (α i)}
Goal: s.pi t₁ ⊆ s.pi t₂ ↔ (∀ i ∈ s, t₁ i ⊆ t₂ i) ∨ s.pi t₁ = ∅
-- Set.univ_pi_subset_univ_pi_iff
theorem univ_pi_subset_univ_pi_iff :
    pi univ t₁ ⊆ pi univ t₂ ↔ (∀ i, t₁ i ⊆ t₂ i) ∨ ∃ i, t₁ i = ∅ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {t₁ t₂ : (i : ι) → Set (α i)}
Goal: Set.univ.pi t₁ ⊆ Set.univ.pi t₂ ↔ (∀ (i : ι), t₁ i ⊆ t₂ i) ∨ ∃ i, t₁ i = ∅
-- Set.eval_preimage
theorem eval_preimage [DecidableEq ι] {s : Set (α i)} :
    eval i ⁻¹' s = pi univ (update (fun i => univ) i s) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {i : ι} [DecidableEq ι] {s : Set (α i)}
Goal: Function.eval i ⁻¹' s = Set.univ.pi (Function.update (fun i => Set.univ) i s)
-- Set.eval_preimage'
theorem eval_preimage' [DecidableEq ι] {s : Set (α i)} :
    eval i ⁻¹' s = pi {i} (update (fun i => univ) i s) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {i : ι} [DecidableEq ι] {s : Set (α i)}
Goal: Function.eval i ⁻¹' s = {i}.pi (Function.update (fun i => Set.univ) i s)
-- Set.update_preimage_pi
theorem update_preimage_pi [DecidableEq ι] {f : ∀ i, α i} (hi : i ∈ s)
    (hf : ∀ j ∈ s, j ≠ i → f j ∈ t j) : update f i ⁻¹' s.pi t = t i 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} [DecidableEq ι] {f : (i : ι) → α i} (hi : i ∈ s) (hf : ∀ j ∈ s, j ≠ i → f j ∈ t j)
Goal: Function.update f i ⁻¹' s.pi t = t i
-- Set.update_image
theorem update_image [DecidableEq ι] (x : (i : ι) → β i) (i : ι) (s : Set (β i)) :
    update x i '' s = Set.univ.pi (update (fun j ↦ {x j}) i s) 
Arguments: {ι : Type u_1} {β : ι → Type u_3} [DecidableEq ι] (x : (i : ι) → β i) (i : ι) (s : Set (β i))
Goal: Function.update x i '' s = Set.univ.pi (Function.update (fun j => {x j}) i s)
-- Set.update_preimage_univ_pi
theorem update_preimage_univ_pi [DecidableEq ι] {f : ∀ i, α i} (hf : ∀ j ≠ i, f j ∈ t j) :
    update f i ⁻¹' pi univ t = t i 
Arguments: {ι : Type u_1} {α : ι → Type u_2} {t : (i : ι) → Set (α i)} {i : ι} [DecidableEq ι] {f : (i : ι) → α i} (hf : ∀ (j : ι), j ≠ i → f j ∈ t j)
Goal: Function.update f i ⁻¹' Set.univ.pi t = t i
-- Set.subset_pi_eval_image
theorem subset_pi_eval_image (s : Set ι) (u : Set (∀ i, α i)) : u ⊆ pi s fun i => eval i '' u 
Arguments: {ι : Type u_1} {α : ι → Type u_2} (s : Set ι) (u : Set ((i : ι) → α i))
Goal: u ⊆ s.pi fun i => Function.eval i '' u
-- Set.univ_pi_ite
theorem univ_pi_ite (s : Set ι) [DecidablePred (· ∈ s)] (t : ∀ i, Set (α i)) :
    (pi univ fun i => if i ∈ s then t i else univ) = s.pi t 
Arguments: {ι : Type u_1} {α : ι → Type u_2} (s : Set ι) [DecidablePred fun x => x ∈ s] (t : (i : ι) → Set (α i))
Goal: (Set.univ.pi fun i => if i ∈ s then t i else Set.univ) = s.pi t
-- Equiv.piCongrLeft_symm_preimage_pi
theorem piCongrLeft_symm_preimage_pi (f : ι' ≃ ι) (s : Set ι') (t : ∀ i, Set (α i)) :
    (f.piCongrLeft α).symm ⁻¹' s.pi (fun i' => t <| f i') = (f '' s).pi t 
Arguments: {ι : Type u_1} {ι' : Type u_2} {α : ι → Type u_3} (f : ι' ≃ ι) (s : Set ι') (t : (i : ι) → Set (α i))
Goal: (⇑(Equiv.piCongrLeft α f).symm ⁻¹' s.pi fun i' => t (f i')) = (⇑f '' s).pi t
-- Equiv.piCongrLeft_symm_preimage_univ_pi
theorem piCongrLeft_symm_preimage_univ_pi (f : ι' ≃ ι) (t : ∀ i, Set (α i)) :
    (f.piCongrLeft α).symm ⁻¹' univ.pi (fun i' => t <| f i') = univ.pi t 
Arguments: {ι : Type u_1} {ι' : Type u_2} {α : ι → Type u_3} (f : ι' ≃ ι) (t : (i : ι) → Set (α i))
Goal: (⇑(Equiv.piCongrLeft α f).symm ⁻¹' Set.univ.pi fun i' => t (f i')) = Set.univ.pi t
-- Equiv.piCongrLeft_preimage_pi
theorem piCongrLeft_preimage_pi (f : ι' ≃ ι) (s : Set ι') (t : ∀ i, Set (α i)) :
    f.piCongrLeft α ⁻¹' (f '' s).pi t = s.pi fun i => t (f i) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {α : ι → Type u_3} (f : ι' ≃ ι) (s : Set ι') (t : (i : ι) → Set (α i))
Goal: ⇑(Equiv.piCongrLeft α f) ⁻¹' (⇑f '' s).pi t = s.pi fun i => t (f i)
-- Equiv.piCongrLeft_preimage_univ_pi
theorem piCongrLeft_preimage_univ_pi (f : ι' ≃ ι) (t : ∀ i, Set (α i)) :
    f.piCongrLeft α ⁻¹' univ.pi t = univ.pi fun i => t (f i) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {α : ι → Type u_3} (f : ι' ≃ ι) (t : (i : ι) → Set (α i))
Goal: ⇑(Equiv.piCongrLeft α f) ⁻¹' Set.univ.pi t = Set.univ.pi fun i => t (f i)
-- Equiv.sumPiEquivProdPi_symm_preimage_univ_pi
theorem sumPiEquivProdPi_symm_preimage_univ_pi (π : ι ⊕ ι' → Type*) (t : ∀ i, Set (π i)) :
    (sumPiEquivProdPi π).symm ⁻¹' univ.pi t =
    univ.pi (fun i => t (.inl i)) ×ˢ univ.pi fun i => t (.inr i) 
Arguments: {ι : Type u_1} {ι' : Type u_2} (π : ι ⊕ ι' → Type u_4) (t : (i : ι ⊕ ι') → Set (π i))
Goal: ⇑(Equiv.sumPiEquivProdPi π).symm ⁻¹' Set.univ.pi t =
  (Set.univ.pi fun i => t (Sum.inl i)) ×ˢ Set.univ.pi fun i => t (Sum.inr i)
-- Set.preimage_empty
@[simp]
theorem preimage_empty : f ⁻¹' ∅ = ∅ 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β}
Goal: f ⁻¹' ∅ = ∅
-- Set.preimage_congr
theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {s : Set β} (h : ∀ (x : α), f x = g x)
Goal: f ⁻¹' s = g ⁻¹' s
-- Set.preimage_mono
@[gcongr]
theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s t : Set β} (h : s ⊆ t)
Goal: f ⁻¹' s ⊆ f ⁻¹' t
-- Set.preimage_univ
@[simp, mfld_simps]
theorem preimage_univ : f ⁻¹' univ = univ 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β}
Goal: f ⁻¹' Set.univ = Set.univ
-- Set.subset_preimage_univ
theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set α}
Goal: s ⊆ f ⁻¹' Set.univ
-- Set.preimage_inter
@[simp, mfld_simps]
theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s t : Set β}
Goal: f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t
-- Set.preimage_union
@[simp]
theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s t : Set β}
Goal: f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t
-- Set.preimage_compl
@[simp]
theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set β}
Goal: f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ
-- Set.preimage_diff
@[simp]
theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (s t : Set β)
Goal: f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t
-- Set.preimage_symmDiff
@[simp]
lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} (s t : Set β)
Goal: f ⁻¹' symmDiff s t = symmDiff (f ⁻¹' s) (f ⁻¹' t)
-- Set.preimage_ite
@[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) 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (s t₁ t₂ : Set β)
Goal: f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂)
-- Set.preimage_setOf_eq
@[simp]
theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a | p a } = { a | p (f a) } 
Arguments: {α : Type u_1} {β : Type u_2} {p : α → Prop} {f : β → α}
Goal: f ⁻¹' {a | p a} = {a | p (f a)}
-- Set.preimage_id_eq
@[simp]
theorem preimage_id_eq : preimage (id : α → α) = id 
Arguments: {α : Type u_1}
Goal: Set.preimage id = id
-- Set.preimage_id
@[mfld_simps]
theorem preimage_id {s : Set α} : id ⁻¹' s = s 
Arguments: {α : Type u_1} {s : Set α}
Goal: id ⁻¹' s = s
-- Set.preimage_id'
@[simp, mfld_simps]
theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s 
Arguments: {α : Type u_1} {s : Set α}
Goal: (fun x => x) ⁻¹' s = s
-- Set.preimage_const_of_mem
@[simp]
theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ 
Arguments: {α : Type u_1} {β : Type u_2} {b : β} {s : Set β} (h : b ∈ s)
Goal: (fun x => b) ⁻¹' s = Set.univ
-- Set.preimage_const_of_not_mem
@[simp]
theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ 
Arguments: {α : Type u_1} {β : Type u_2} {b : β} {s : Set β} (h : b ∉ s)
Goal: (fun x => b) ⁻¹' s = ∅
-- Set.preimage_const
theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
    (fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ 
Arguments: {α : Type u_1} {β : Type u_2} (b : β) (s : Set β) [Decidable (b ∈ s)]
Goal: (fun x => b) ⁻¹' s = if b ∈ s then Set.univ else ∅
-- Set.exists_eq_const_of_preimage_singleton
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 
Arguments: {α : Type u_1} {β : Type u_2} [Nonempty β] {f : α → β} (hf : ∀ (b : β), f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = Set.univ)
Goal: ∃ b, f = Function.const α b
-- Set.preimage_comp
theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : β → γ} {s : Set γ}
Goal: g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s)
-- Set.preimage_comp_eq
theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : β → γ}
Goal: Set.preimage (g ∘ f) = Set.preimage f ∘ Set.preimage g
-- Set.preimage_iterate_eq
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] 
Arguments: {α : Type u_1} {f : α → α} {n : ℕ}
Goal: Set.preimage f^[n] = (Set.preimage f)^[n]
-- Set.preimage_preimage
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
    f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {g : β → γ} {f : α → β} {s : Set γ}
Goal: f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s
-- Set.eq_preimage_subtype_val_iff
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 
Arguments: {α : Type u_1} {p : α → Prop} {s : Set (Subtype p)} {t : Set α}
Goal: s = Subtype.val ⁻¹' t ↔ ∀ (x : α) (h : p x), ⟨x, h⟩ ∈ s ↔ x ∈ t
-- Set.nonempty_of_nonempty_preimage
theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) :
    s.Nonempty 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty)
Goal: s.Nonempty
-- Set.preimage_singleton_true
@[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a | p a} 
Arguments: {α : Type u_1} (p : α → Prop)
Goal: p ⁻¹' {True} = {a | p a}
-- Set.preimage_singleton_false
@[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a | ¬p a} 
Arguments: {α : Type u_1} (p : α → Prop)
Goal: p ⁻¹' {False} = {a | ¬p a}
-- Set.preimage_subtype_coe_eq_compl
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 
Arguments: {α : Type u_1} {s u v : Set α} (hsuv : s ⊆ u ∪ v) (H : s ∩ (u ∩ v) = ∅)
Goal: Subtype.val ⁻¹' u = (Subtype.val ⁻¹' v)ᶜ
-- Set.mem_image_iff_bex
@[deprecated mem_image (since := "2024-03-23")]
theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} :
    y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set α} {y : β}
Goal: y ∈ f '' s ↔ ∃ x, ∃ (_ : x ∈ s), f x = y
-- Set.image_eta
theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} (f : α → β)
Goal: f '' s = (fun x => f x) '' s
-- Function.Injective.mem_set_image
theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} :
    f a ∈ f '' s ↔ a ∈ s 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) {s : Set α} {a : α}
Goal: f a ∈ f '' s ↔ a ∈ s
-- Set.preimage_subset_of_surjOn
lemma preimage_subset_of_surjOn {t : Set β} (hf : Injective f) (h : SurjOn f s t) :
    f ⁻¹' t ⊆ s 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set α} {t : Set β} (hf : Function.Injective f) (h : Set.SurjOn f s t)
Goal: f ⁻¹' t ⊆ s
-- Set.forall_mem_image
theorem forall_mem_image {f : α → β} {s : Set α} {p : β → Prop} :
    (∀ y ∈ f '' s, p y) ↔ ∀ ⦃x⦄, x ∈ s → p (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set α} {p : β → Prop}
Goal: (∀ y ∈ f '' s, p y) ↔ ∀ ⦃x : α⦄, x ∈ s → p (f x)
-- Set.exists_mem_image
theorem exists_mem_image {f : α → β} {s : Set α} {p : β → Prop} :
    (∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set α} {p : β → Prop}
Goal: (∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x)
-- Set.mem_image_elim
@[deprecated forall_mem_image (since := "2024-02-21")]
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 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x ∈ s, C (f x)) {y : β}
Goal: y ∈ f '' s → C y
-- Set.mem_image_elim_on
@[deprecated forall_mem_image (since := "2024-02-21")]
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 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s) (h : ∀ x ∈ s, C (f x))
Goal: C y
-- Set.image_congr
@[congr]
theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a)
Goal: f '' s = g '' s
-- Set.image_congr'
theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {s : Set α} (h : ∀ (x : α), f x = g x)
Goal: f '' s = g '' s
-- Set.image_mono
@[gcongr]
lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s t : Set α} (h : s ⊆ t)
Goal: f '' s ⊆ f '' t
-- Set.image_comp
theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : β → γ) (g : α → β) (a : Set α)
Goal: f ∘ g '' a = f '' (g '' a)
-- Set.image_comp_eq
theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : β → γ}
Goal: Set.image (g ∘ f) = Set.image g ∘ Set.image f
-- Set.image_image
theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (g : β → γ) (f : α → β) (s : Set α)
Goal: g '' (f '' s) = (fun x => g (f x)) '' s
-- Set.image_comm
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' 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {β' : Type u_6} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ} (h_comm : ∀ (a : α), f (g a) = g' (f' a))
Goal: f '' (g '' s) = g' '' (f' '' s)
-- Function.Semiconj.set_image
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) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α} {gb : β → β} (h : Function.Semiconj f ga gb)
Goal: Function.Semiconj (Set.image f) (Set.image ga) (Set.image gb)
-- Function.Commute.set_image
theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) :
    Function.Commute (image f) (image g) 
Arguments: {α : Type u_1} {f g : α → α} (h : Function.Commute f g)
Goal: Function.Commute (Set.image f) (Set.image g)
-- Set.image_subset
@[gcongr]
theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b 
Arguments: {α : Type u_1} {β : Type u_2} {a b : Set α} (f : α → β) (h : a ⊆ b)
Goal: f '' a ⊆ f '' b
-- Set.monotone_image
lemma monotone_image {f : α → β} : Monotone (image f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β}
Goal: Monotone (Set.image f)
-- Set.image_union
theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (s t : Set α)
Goal: f '' (s ∪ t) = f '' s ∪ f '' t
-- Set.image_empty
@[simp]
theorem image_empty (f : α → β) : f '' ∅ = ∅ 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β)
Goal: f '' ∅ = ∅
-- Set.image_inter_subset
theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (s t : Set α)
Goal: f '' (s ∩ t) ⊆ f '' s ∩ f '' t
-- Set.image_inter_on
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 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y)
Goal: f '' (s ∩ t) = f '' s ∩ f '' t
-- Set.image_inter
theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s t : Set α} (H : Function.Injective f)
Goal: f '' (s ∩ t) = f '' s ∩ f '' t
-- Set.image_univ_of_surjective
theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ 
Arguments: {β : Type u_2} {ι : Type u_6} {f : ι → β} (H : Function.Surjective f)
Goal: f '' Set.univ = Set.univ
-- Set.image_singleton
@[simp]
theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {a : α}
Goal: f '' {a} = {f a}
-- Set.Nonempty.image_const
@[simp]
theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} (hs : s.Nonempty) (a : β)
Goal: (fun x => a) '' s = {a}
-- Set.image_eq_empty
@[simp, mfld_simps]
theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ 
Arguments: {α : Type u_6} {β : Type u_7} {f : α → β} {s : Set α}
Goal: f '' s = ∅ ↔ s = ∅
-- Set.preimage_compl_eq_image_compl
theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) :
    HasCompl.compl ⁻¹' S = HasCompl.compl '' S 
Arguments: {α : Type u_1} [BooleanAlgebra α] (S : Set α)
Goal: compl ⁻¹' S = compl '' S
-- Set.mem_compl_image
theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
    t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S 
Arguments: {α : Type u_1} [BooleanAlgebra α] (t : α) (S : Set α)
Goal: t ∈ compl '' S ↔ tᶜ ∈ S
-- Set.image_id_eq
@[simp]
theorem image_id_eq : image (id : α → α) = id 
Arguments: {α : Type u_1}
Goal: Set.image id = id
-- Set.image_id'
@[simp]
theorem image_id' (s : Set α) : (fun x => x) '' s = s 
Arguments: {α : Type u_1} (s : Set α)
Goal: (fun x => x) '' s = s
-- Set.image_id
theorem image_id (s : Set α) : id '' s = s 
Arguments: {α : Type u_1} (s : Set α)
Goal: id '' s = s
-- Set.image_iterate_eq
lemma image_iterate_eq {f : α → α} {n : ℕ} : image (f^[n]) = (image f)^[n] 
Arguments: {α : Type u_1} {f : α → α} {n : ℕ}
Goal: Set.image f^[n] = (Set.image f)^[n]
-- Set.compl_compl_image
theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
    HasCompl.compl '' (HasCompl.compl '' S) = S 
Arguments: {α : Type u_1} [BooleanAlgebra α] (S : Set α)
Goal: compl '' (compl '' S) = S
-- Set.image_insert_eq
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
    f '' insert a s = insert (f a) (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {a : α} {s : Set α}
Goal: f '' insert a s = insert (f a) (f '' s)
-- Set.image_pair
theorem image_pair (f : α → β) (a b : α) : f '' {a, b} = {f a, f b} 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (a b : α)
Goal: f '' {a, b} = {f a, f b}
-- Set.image_subset_preimage_of_inverse
theorem image_subset_preimage_of_inverse {f : α → β} {g : β → α} (I : LeftInverse g f) (s : Set α) :
    f '' s ⊆ g ⁻¹' s 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {g : β → α} (I : Function.LeftInverse g f) (s : Set α)
Goal: f '' s ⊆ g ⁻¹' s
-- Set.preimage_subset_image_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 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {g : β → α} (I : Function.LeftInverse g f) (s : Set β)
Goal: f ⁻¹' s ⊆ g '' s
-- Set.image_eq_preimage_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 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {g : β → α} (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f)
Goal: Set.image f = Set.preimage g