ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.biUnion_iUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
    ⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} (s : ι → Set α) (t : α → Set β)
Goal: ⋃ x ∈ ⋃ i, s i, t x = ⋃ i, ⋃ x ∈ s i, t x
-- Set.biInter_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
    ⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} (s : ι → Set α) (t : α → Set β)
Goal: ⋂ x ∈ ⋃ i, s i, t x = ⋂ i, ⋂ x ∈ s i, t x
-- Set.sUnion_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i 
Arguments: {α : Type u_1} {ι : Sort u_4} (s : ι → Set (Set α))
Goal: ⋃₀ ⋃ i, s i = ⋃ i, ⋃₀ s i
-- Set.sInter_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i 
Arguments: {α : Type u_1} {ι : Sort u_4} (s : ι → Set (Set α))
Goal: ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i
-- Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
    (hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C 
Arguments: {α : Type u_11} {β : Type u_12} (C : Set (Set α)) {f : (s : ↑C) → β → ↑↑s} (hf : ∀ (s : ↑C), Function.Surjective (f s))
Goal: (⋃ y, Set.range fun s => ↑(f s y)) = ⋃₀ C
-- Set.iUnion_range_eq_iUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
    (hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} (C : ι → Set α) {f : (x : ι) → β → ↑(C x)} (hf : ∀ (x : ι), Function.Surjective (f x))
Goal: (⋃ y, Set.range fun x => ↑(f x y)) = ⋃ x, C x
-- Set.union_distrib_iInter_left
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i 
Arguments: {α : Type u_1} {ι : Sort u_4} (s : ι → Set α) (t : Set α)
Goal: t ∪ ⋂ i, s i = ⋂ i, t ∪ s i
-- Set.union_distrib_iInter₂_left
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
    (s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} (s : Set α) (t : (i : ι) → κ i → Set α)
Goal: s ∪ ⋂ i, ⋂ j, t i j = ⋂ i, ⋂ j, s ∪ t i j
-- Set.union_distrib_iInter_right
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t 
Arguments: {α : Type u_1} {ι : Sort u_4} (s : ι → Set α) (t : Set α)
Goal: (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t
-- Set.union_distrib_iInter₂_right
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
    (⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} (s : (i : ι) → κ i → Set α) (t : Set α)
Goal: (⋂ i, ⋂ j, s i j) ∪ t = ⋂ i, ⋂ j, s i j ∪ t
-- Set.mapsTo_sUnion
@[simp]
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} :
    MapsTo f (⋃₀ S) t ↔ ∀ s ∈ S, MapsTo f s t 
Arguments: {α : Type u_1} {β : Type u_2} {S : Set (Set α)} {t : Set β} {f : α → β}
Goal: Set.MapsTo f (⋃₀ S) t ↔ ∀ s ∈ S, Set.MapsTo f s t
-- Set.mapsTo_iUnion
@[simp]
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} :
    MapsTo f (⋃ i, s i) t ↔ ∀ i, MapsTo f (s i) t 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : ι → Set α} {t : Set β} {f : α → β}
Goal: Set.MapsTo f (⋃ i, s i) t ↔ ∀ (i : ι), Set.MapsTo f (s i) t
-- Set.mapsTo_iUnion₂
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β} :
    MapsTo f (⋃ (i) (j), s i j) t ↔ ∀ i j, MapsTo f (s i j) t 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} {s : (i : ι) → κ i → Set α} {t : Set β} {f : α → β}
Goal: Set.MapsTo f (⋃ i, ⋃ j, s i j) t ↔ ∀ (i : ι) (j : κ i), Set.MapsTo f (s i j) t
-- Set.mapsTo_iUnion_iUnion
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
    (H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ (i : ι), Set.MapsTo f (s i) (t i))
Goal: Set.MapsTo f (⋃ i, s i) (⋃ i, t i)
-- Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
    (H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} {s : (i : ι) → κ i → Set α} {t : (i : ι) → κ i → Set β} {f : α → β} (H : ∀ (i : ι) (j : κ i), Set.MapsTo f (s i j) (t i j))
Goal: Set.MapsTo f (⋃ i, ⋃ j, s i j) (⋃ i, ⋃ j, t i j)
-- Set.mapsTo_sInter
@[simp]
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} :
    MapsTo f s (⋂₀ T) ↔ ∀ t ∈ T, MapsTo f s t 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {T : Set (Set β)} {f : α → β}
Goal: Set.MapsTo f s (⋂₀ T) ↔ ∀ t ∈ T, Set.MapsTo f s t
-- Set.mapsTo_iInter
@[simp]
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} :
    MapsTo f s (⋂ i, t i) ↔ ∀ i, MapsTo f s (t i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : Set α} {t : ι → Set β} {f : α → β}
Goal: Set.MapsTo f s (⋂ i, t i) ↔ ∀ (i : ι), Set.MapsTo f s (t i)
-- Set.mapsTo_iInter₂
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β} :
    MapsTo f s (⋂ (i) (j), t i j) ↔ ∀ i j, MapsTo f s (t i j) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} {s : Set α} {t : (i : ι) → κ i → Set β} {f : α → β}
Goal: Set.MapsTo f s (⋂ i, ⋂ j, t i j) ↔ ∀ (i : ι) (j : κ i), Set.MapsTo f s (t i j)
-- Set.mapsTo_iInter_iInter
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
    (H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ (i : ι), Set.MapsTo f (s i) (t i))
Goal: Set.MapsTo f (⋂ i, s i) (⋂ i, t i)
-- Set.mapsTo_iInter₂_iInter₂
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
    (H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} {s : (i : ι) → κ i → Set α} {t : (i : ι) → κ i → Set β} {f : α → β} (H : ∀ (i : ι) (j : κ i), Set.MapsTo f (s i j) (t i j))
Goal: Set.MapsTo f (⋂ i, ⋂ j, s i j) (⋂ i, ⋂ j, t i j)
-- Set.image_iInter_subset
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} (s : ι → Set α) (f : α → β)
Goal: f '' ⋂ i, s i ⊆ ⋂ i, f '' s i
-- Set.image_iInter₂_subset
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
    (f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} (s : (i : ι) → κ i → Set α) (f : α → β)
Goal: f '' ⋂ i, ⋂ j, s i j ⊆ ⋂ i, ⋂ j, f '' s i j
-- Set.image_sInter_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s 
Arguments: {α : Type u_1} {β : Type u_2} (S : Set (Set α)) (f : α → β)
Goal: f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s
-- Set.injective_iff_injective_of_iUnion_eq_univ
theorem injective_iff_injective_of_iUnion_eq_univ :
    Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : α → β} {U : ι → Set β} (hU : Set.iUnion U = Set.univ)
Goal: Function.Injective f ↔ ∀ (i : ι), Function.Injective ((U i).restrictPreimage f)
-- Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
    Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : α → β} {U : ι → Set β} (hU : Set.iUnion U = Set.univ)
Goal: Function.Surjective f ↔ ∀ (i : ι), Function.Surjective ((U i).restrictPreimage f)
-- Set.bijective_iff_bijective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
    Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : α → β} {U : ι → Set β} (hU : Set.iUnion U = Set.univ)
Goal: Function.Bijective f ↔ ∀ (i : ι), Function.Bijective ((U i).restrictPreimage f)
-- Set.InjOn.image_iInter_eq
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
    (f '' ⋂ i, s i) = ⋂ i, f '' s i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} [Nonempty ι] {s : ι → Set α} {f : α → β} (h : Set.InjOn f (⋃ i, s i))
Goal: f '' ⋂ i, s i = ⋂ i, f '' s i
-- Set.InjOn.image_biInter_eq
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
    {f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
    (f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {p : ι → Prop} {s : (i : ι) → p i → Set α} (hp : ∃ i, p i) {f : α → β} (h : Set.InjOn f (⋃ i, ⋃ (hi : p i), s i hi))
Goal: f '' ⋂ i, ⋂ (hi : p i), s i hi = ⋂ i, ⋂ (hi : p i), f '' s i hi
-- Set.image_iInter
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
    (f '' ⋂ i, s i) = ⋂ i, f '' s i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : α → β} (hf : Function.Bijective f) (s : ι → Set α)
Goal: f '' ⋂ i, s i = ⋂ i, f '' s i
-- Set.image_iInter₂
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
    (f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} {f : α → β} (hf : Function.Bijective f) (s : (i : ι) → κ i → Set α)
Goal: f '' ⋂ i, ⋂ j, s i j = ⋂ i, ⋂ j, f '' s i j
-- Set.inj_on_iUnion_of_directed
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
    (hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : ι → Set α} (hs : Directed (fun x x_1 => x ⊆ x_1) s) {f : α → β} (hf : ∀ (i : ι), Set.InjOn f (s i))
Goal: Set.InjOn f (⋃ i, s i)
-- Set.surjOn_sUnion
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
    SurjOn f s (⋃₀T) 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, Set.SurjOn f s t)
Goal: Set.SurjOn f s (⋃₀ T)
-- Set.surjOn_iUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
    SurjOn f s (⋃ i, t i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ (i : ι), Set.SurjOn f s (t i))
Goal: Set.SurjOn f s (⋃ i, t i)
-- Set.surjOn_iUnion_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
    (H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ (i : ι), Set.SurjOn f (s i) (t i))
Goal: Set.SurjOn f (⋃ i, s i) (⋃ i, t i)
-- Set.surjOn_iUnion₂
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
    (H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} {s : Set α} {t : (i : ι) → κ i → Set β} {f : α → β} (H : ∀ (i : ι) (j : κ i), Set.SurjOn f s (t i j))
Goal: Set.SurjOn f s (⋃ i, ⋃ j, t i j)
-- Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
    (H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} {s : (i : ι) → κ i → Set α} {t : (i : ι) → κ i → Set β} {f : α → β} (H : ∀ (i : ι) (j : κ i), Set.SurjOn f (s i j) (t i j))
Goal: Set.SurjOn f (⋃ i, ⋃ j, s i j) (⋃ i, ⋃ j, t i j)
-- Set.surjOn_iInter
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
    (H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ (i : ι), Set.SurjOn f (s i) t) (Hinj : Set.InjOn f (⋃ i, s i))
Goal: Set.SurjOn f (⋂ i, s i) t
-- Set.surjOn_iInter_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
    (H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ (i : ι), Set.SurjOn f (s i) (t i)) (Hinj : Set.InjOn f (⋃ i, s i))
Goal: Set.SurjOn f (⋂ i, s i) (⋂ i, t i)
-- Set.bijOn_iUnion
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
    (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ (i : ι), Set.BijOn f (s i) (t i)) (Hinj : Set.InjOn f (⋃ i, s i))
Goal: Set.BijOn f (⋃ i, s i) (⋃ i, t i)
-- Set.bijOn_iInter
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
    (H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ (i : ι), Set.BijOn f (s i) (t i)) (Hinj : Set.InjOn f (⋃ i, s i))
Goal: Set.BijOn f (⋂ i, s i) (⋂ i, t i)
-- Set.bijOn_iUnion_of_directed
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
    {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : ι → Set α} (hs : Directed (fun x x_1 => x ⊆ x_1) s) {t : ι → Set β} {f : α → β} (H : ∀ (i : ι), Set.BijOn f (s i) (t i))
Goal: Set.BijOn f (⋃ i, s i) (⋃ i, t i)
-- Set.bijOn_iInter_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
    {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} [Nonempty ι] {s : ι → Set α} (hs : Directed (fun x x_1 => x ⊆ x_1) s) {t : ι → Set β} {f : α → β} (H : ∀ (i : ι), Set.BijOn f (s i) (t i))
Goal: Set.BijOn f (⋂ i, s i) (⋂ i, t i)
-- Set.image_iUnion
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : α → β} {s : ι → Set α}
Goal: f '' ⋃ i, s i = ⋃ i, f '' s i
-- Set.image_iUnion₂
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
    (f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} (f : α → β) (s : (i : ι) → κ i → Set α)
Goal: f '' ⋃ i, ⋃ j, s i j = ⋃ i, ⋃ j, f '' s i j
-- Set.univ_subtype
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} 
Arguments: {α : Type u_1} {p : α → Prop}
Goal: Set.univ = ⋃ x, ⋃ (h : p x), {⟨x, h⟩}
-- Set.range_eq_iUnion
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} 
Arguments: {α : Type u_1} {ι : Sort u_11} (f : ι → α)
Goal: Set.range f = ⋃ i, {f i}
-- Set.image_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set α)
Goal: f '' s = ⋃ i ∈ s, {f i}
-- Set.biUnion_range
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : ι → α} {g : α → Set β}
Goal: ⋃ x ∈ Set.range f, g x = ⋃ y, g (f y)
-- Set.iUnion_iUnion_eq'
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
    ⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : ι → α} {g : α → Set β}
Goal: ⋃ x, ⋃ y, ⋃ (_ : f y = x), g x = ⋃ y, g (f y)
-- Set.biInter_range
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : ι → α} {g : α → Set β}
Goal: ⋂ x ∈ Set.range f, g x = ⋂ y, g (f y)
-- Set.iInter_iInter_eq'
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
    ⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : ι → α} {g : α → Set β}
Goal: ⋂ x, ⋂ y, ⋂ (_ : f y = x), g x = ⋂ y, g (f y)
-- Set.biUnion_image
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set γ} {f : γ → α} {g : α → Set β}
Goal: ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y)
-- Set.biInter_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set γ} {f : γ → α} {g : α → Set β}
Goal: ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y)
-- Set.monotone_preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β}
Goal: Monotone (Set.preimage f)
-- Set.preimage_iUnion
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : α → β} {s : ι → Set β}
Goal: f ⁻¹' ⋃ i, s i = ⋃ i, f ⁻¹' s i
-- Set.preimage_iUnion₂
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
    (f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} {f : α → β} {s : (i : ι) → κ i → Set β}
Goal: f ⁻¹' ⋃ i, ⋃ j, s i j = ⋃ i, ⋃ j, f ⁻¹' s i j
-- Set.image_sUnion
theorem image_sUnion {f : α → β} {s : Set (Set α)} : (f '' ⋃₀ s) = ⋃₀ (image f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set (Set α)}
Goal: f '' ⋃₀ s = ⋃₀ (Set.image f '' s)
-- Set.preimage_sUnion
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set (Set β)}
Goal: f ⁻¹' ⋃₀ s = ⋃ t ∈ s, f ⁻¹' t
-- Set.preimage_iInter
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : α → β} {s : ι → Set β}
Goal: f ⁻¹' ⋂ i, s i = ⋂ i, f ⁻¹' s i
-- Set.preimage_iInter₂
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
    (f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} {f : α → β} {s : (i : ι) → κ i → Set β}
Goal: f ⁻¹' ⋂ i, ⋂ j, s i j = ⋂ i, ⋂ j, f ⁻¹' s i j
-- Set.preimage_sInter
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set (Set β)}
Goal: f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t
-- Set.biUnion_preimage_singleton
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set β)
Goal: ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s
-- Set.biUnion_range_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β)
Goal: ⋃ y ∈ Set.range f, f ⁻¹' {y} = Set.univ
-- Set.prod_iUnion
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : Set α} {t : ι → Set β}
Goal: s ×ˢ ⋃ i, t i = ⋃ i, s ×ˢ t i
-- Set.prod_iUnion₂
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
    (s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} {s : Set α} {t : (i : ι) → κ i → Set β}
Goal: s ×ˢ ⋃ i, ⋃ j, t i j = ⋃ i, ⋃ j, s ×ˢ t i j
-- Set.prod_sUnion
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {C : Set (Set β)}
Goal: s ×ˢ ⋃₀ C = ⋃₀ ((fun t => s ×ˢ t) '' C)
-- Set.iUnion_prod_const
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : ι → Set α} {t : Set β}
Goal: (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t
-- Set.iUnion₂_prod_const
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
    (⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {κ : ι → Sort u_7} {s : (i : ι) → κ i → Set α} {t : Set β}
Goal: (⋃ i, ⋃ j, s i j) ×ˢ t = ⋃ i, ⋃ j, s i j ×ˢ t
-- Set.sUnion_prod_const
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
    ⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) 
Arguments: {α : Type u_1} {β : Type u_2} {C : Set (Set α)} {t : Set β}
Goal: ⋃₀ C ×ˢ t = ⋃₀ ((fun s => s ×ˢ t) '' C)
-- Set.iUnion_prod
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
    ⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i 
Arguments: {ι : Type u_11} {ι' : Type u_12} {α : Type u_13} {β : Type u_14} (s : ι → Set α) (t : ι' → Set β)
Goal: ⋃ x, s x.1 ×ˢ t x.2 = (⋃ i, s i) ×ˢ ⋃ i, t i
-- Set.iUnion_prod'
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : β × γ → Set α)
Goal: ⋃ x, f x = ⋃ i, ⋃ j, f (i, j)
-- Set.iUnion_prod_of_monotone
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
    (ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s) (ht : Monotone t)
Goal: ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x
-- Set.sInter_prod_sInter_subset
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
    ⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 
Arguments: {α : Type u_1} {β : Type u_2} (S : Set (Set α)) (T : Set (Set β))
Goal: ⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2
-- Set.sInter_prod_sInter
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
    ⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 
Arguments: {α : Type u_1} {β : Type u_2} {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty)
Goal: ⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2
-- Set.sInter_prod
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
    ⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t 
Arguments: {α : Type u_1} {β : Type u_2} {S : Set (Set α)} (hS : S.Nonempty) (t : Set β)
Goal: ⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t
-- Set.prod_sInter
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
    s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t 
Arguments: {α : Type u_1} {β : Type u_2} {T : Set (Set β)} (hT : T.Nonempty) (s : Set α)
Goal: s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t
-- Set.prod_iInter
theorem prod_iInter {s : Set α} {t : ι → Set β} [hι : Nonempty ι] :
    (s ×ˢ ⋂ i, t i) = ⋂ i, s ×ˢ t i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {s : Set α} {t : ι → Set β} [hι : Nonempty ι]
Goal: s ×ˢ ⋂ i, t i = ⋂ i, s ×ˢ t i
-- Set.image2_eq_iUnion
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (s : Set α) (t : Set β)
Goal: Set.image2 f s t = ⋃ i ∈ s, ⋃ j ∈ t, {f i j}
-- Set.iUnion_image_left
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) {s : Set α} {t : Set β}
Goal: ⋃ a ∈ s, f a '' t = Set.image2 f s t
-- Set.iUnion_image_right
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) {s : Set α} {t : Set β}
Goal: ⋃ b ∈ t, (fun x => f x b) '' s = Set.image2 f s t
-- Set.image2_iUnion_left
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
    image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {ι : Sort u_4} (f : α → β → γ) (s : ι → Set α) (t : Set β)
Goal: Set.image2 f (⋃ i, s i) t = ⋃ i, Set.image2 f (s i) t
-- Set.image2_iUnion_right
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
    image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {ι : Sort u_4} (f : α → β → γ) (s : Set α) (t : ι → Set β)
Goal: Set.image2 f s (⋃ i, t i) = ⋃ i, Set.image2 f s (t i)