ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.biInter_and
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
    ⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
      ⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} (p : ι → Prop) (q : ι → ι' → Prop) (s : (x : ι) → (y : ι') → p x ∧ q x y → Set α)
Goal: ⋂ x, ⋂ y, ⋂ (h : p x ∧ q x y), s x y h = ⋂ x, ⋂ (hx : p x), ⋂ y, ⋂ (hy : q x y), s x y ⋯
-- Set.biInter_and'
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
    ⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
      ⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} (p : ι' → Prop) (q : ι → ι' → Prop) (s : (x : ι) → (y : ι') → p y ∧ q x y → Set α)
Goal: ⋂ x, ⋂ y, ⋂ (h : p y ∧ q x y), s x y h = ⋂ y, ⋂ (hy : p y), ⋂ x, ⋂ (hx : q x y), s x y ⋯
-- Set.iUnion_iUnion_eq_or_left
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
    ⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) 
Arguments: {α : Type u_1} {β : Type u_2} {b : β} {p : β → Prop} {s : (x : β) → x = b ∨ p x → Set α}
Goal: ⋃ x, ⋃ (h : x = b ∨ p x), s x h = s b ⋯ ∪ ⋃ x, ⋃ (h : p x), s x ⋯
-- Set.iInter_iInter_eq_or_left
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
    ⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) 
Arguments: {α : Type u_1} {β : Type u_2} {b : β} {p : β → Prop} {s : (x : β) → x = b ∨ p x → Set α}
Goal: ⋂ x, ⋂ (h : x = b ∨ p x), s x h = s b ⋯ ∩ ⋂ x, ⋂ (h : p x), s x ⋯
-- Set.mem_biUnion
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
    y ∈ ⋃ x ∈ s, t x 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x)
Goal: y ∈ ⋃ x ∈ s, t x
-- Set.mem_biInter
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
    y ∈ ⋂ x ∈ s, t x 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x)
Goal: y ∈ ⋂ x ∈ s, t x
-- Set.subset_biUnion_of_mem
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
    u x ⊆ ⋃ x ∈ s, u x 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s)
Goal: u x ⊆ ⋃ x ∈ s, u x
-- Set.biInter_subset_of_mem
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
    ⋂ x ∈ s, t x ⊆ t x 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s)
Goal: ⋂ x ∈ s, t x ⊆ t x
-- Set.biUnion_subset_biUnion_left
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
    ⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x 
Arguments: {α : Type u_1} {β : Type u_2} {s s' : Set α} {t : α → Set β} (h : s ⊆ s')
Goal: ⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x
-- Set.biInter_subset_biInter_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
    ⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x 
Arguments: {α : Type u_1} {β : Type u_2} {s s' : Set α} {t : α → Set β} (h : s' ⊆ s)
Goal: ⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x
-- Set.biUnion_mono
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
    ⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x 
Arguments: {α : Type u_1} {β : Type u_2} {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x)
Goal: ⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x
-- Set.biInter_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
    ⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x 
Arguments: {α : Type u_1} {β : Type u_2} {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x)
Goal: ⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x
-- Set.biUnion_eq_iUnion
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
    ⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 
Arguments: {α : Type u_1} {β : Type u_2} (s : Set α) (t : (x : α) → x ∈ s → Set β)
Goal: ⋃ x, ⋃ (h : x ∈ s), t x h = ⋃ x, t ↑x ⋯
-- Set.biInter_eq_iInter
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
    ⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 
Arguments: {α : Type u_1} {β : Type u_2} (s : Set α) (t : (x : α) → x ∈ s → Set β)
Goal: ⋂ x, ⋂ (h : x ∈ s), t x h = ⋂ x, t ↑x ⋯
-- Set.iUnion_subtype
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
    ⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ 
Arguments: {α : Type u_1} {β : Type u_2} (p : α → Prop) (s : { x // p x } → Set β)
Goal: ⋃ x, s x = ⋃ x, ⋃ (hx : p x), s ⟨x, hx⟩
-- Set.iInter_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
    ⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ 
Arguments: {α : Type u_1} {β : Type u_2} (p : α → Prop) (s : { x // p x } → Set β)
Goal: ⋂ x, s x = ⋂ x, ⋂ (hx : p x), s ⟨x, hx⟩
-- Set.biInter_empty
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ 
Arguments: {α : Type u_1} {β : Type u_2} (u : α → Set β)
Goal: ⋂ x ∈ ∅, u x = Set.univ
-- Set.biInter_univ
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x 
Arguments: {α : Type u_1} {β : Type u_2} (u : α → Set β)
Goal: ⋂ x ∈ Set.univ, u x = ⋂ x, u x
-- Set.biUnion_self
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s 
Arguments: {α : Type u_1} (s : Set α)
Goal: ⋃ x ∈ s, s = s
-- Set.iUnion_nonempty_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s 
Arguments: {α : Type u_1} (s : Set α)
Goal: ⋃ (_ : s.Nonempty), s = s
-- Set.biInter_singleton
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a 
Arguments: {α : Type u_1} {β : Type u_2} (a : α) (s : α → Set β)
Goal: ⋂ x ∈ {a}, s x = s a
-- Set.biInter_union
theorem biInter_union (s t : Set α) (u : α → Set β) :
    ⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x 
Arguments: {α : Type u_1} {β : Type u_2} (s t : Set α) (u : α → Set β)
Goal: ⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x
-- Set.biInter_insert
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
    ⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x 
Arguments: {α : Type u_1} {β : Type u_2} (a : α) (s : Set α) (t : α → Set β)
Goal: ⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x
-- Set.biInter_pair
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b 
Arguments: {α : Type u_1} {β : Type u_2} (a b : α) (s : α → Set β)
Goal: ⋂ x ∈ {a, b}, s x = s a ∩ s b
-- Set.biInter_inter
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
    ⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t 
Arguments: {ι : Type u_11} {α : Type u_12} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α)
Goal: ⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t
-- Set.inter_biInter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
    ⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i 
Arguments: {ι : Type u_11} {α : Type u_12} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α)
Goal: ⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i
-- Set.biUnion_empty
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ 
Arguments: {α : Type u_1} {β : Type u_2} (s : α → Set β)
Goal: ⋃ x ∈ ∅, s x = ∅
-- Set.biUnion_univ
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x 
Arguments: {α : Type u_1} {β : Type u_2} (s : α → Set β)
Goal: ⋃ x ∈ Set.univ, s x = ⋃ x, s x
-- Set.biUnion_singleton
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a 
Arguments: {α : Type u_1} {β : Type u_2} (a : α) (s : α → Set β)
Goal: ⋃ x ∈ {a}, s x = s a
-- Set.biUnion_of_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s 
Arguments: {α : Type u_1} (s : Set α)
Goal: ⋃ x ∈ s, {x} = s
-- Set.biUnion_union
theorem biUnion_union (s t : Set α) (u : α → Set β) :
    ⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x 
Arguments: {α : Type u_1} {β : Type u_2} (s t : Set α) (u : α → Set β)
Goal: ⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x
-- Set.iUnion_coe_set
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
    ⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ 
Arguments: {α : Type u_11} {β : Type u_12} (s : Set α) (f : ↑s → Set β)
Goal: ⋃ i, f i = ⋃ i, ⋃ (h : i ∈ s), f ⟨i, h⟩
-- Set.iInter_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
    ⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ 
Arguments: {α : Type u_11} {β : Type u_12} (s : Set α) (f : ↑s → Set β)
Goal: ⋂ i, f i = ⋂ i, ⋂ (h : i ∈ s), f ⟨i, h⟩
-- Set.biUnion_insert
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
    ⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x 
Arguments: {α : Type u_1} {β : Type u_2} (a : α) (s : Set α) (t : α → Set β)
Goal: ⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x
-- Set.biUnion_pair
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b 
Arguments: {α : Type u_1} {β : Type u_2} (a b : α) (s : α → Set β)
Goal: ⋃ x ∈ {a, b}, s x = s a ∪ s b
-- Set.inter_iUnion₂
theorem inter_iUnion₂ (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.iUnion₂_inter
theorem iUnion₂_inter (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.union_iInter₂
theorem union_iInter₂ (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.iInter₂_union
theorem iInter₂_union (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.mem_sUnion_of_mem
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
    x ∈ ⋃₀S 
Arguments: {α : Type u_1} {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S)
Goal: x ∈ ⋃₀ S
-- Set.not_mem_of_not_mem_sUnion
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
    (ht : t ∈ S) : x ∉ t 
Arguments: {α : Type u_1} {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀ S) (ht : t ∈ S)
Goal: x ∉ t
-- Set.sInter_subset_of_mem
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t 
Arguments: {α : Type u_1} {S : Set (Set α)} {t : Set α} (tS : t ∈ S)
Goal: ⋂₀ S ⊆ t
-- Set.subset_sUnion_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S 
Arguments: {α : Type u_1} {S : Set (Set α)} {t : Set α} (tS : t ∈ S)
Goal: t ⊆ ⋃₀ S
-- Set.subset_sUnion_of_subset
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
    (h₂ : u ∈ t) : s ⊆ ⋃₀t 
Arguments: {α : Type u_1} {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u) (h₂ : u ∈ t)
Goal: s ⊆ ⋃₀ t
-- Set.sUnion_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t 
Arguments: {α : Type u_1} {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t)
Goal: ⋃₀ S ⊆ t
-- Set.sUnion_subset_iff
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t 
Arguments: {α : Type u_1} {s : Set (Set α)} {t : Set α}
Goal: ⋃₀ s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t
-- Set.sUnion_mono_subsets
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
    ⋃₀ s ⊆ ⋃₀ (f '' s) 
Arguments: {α : Type u_1} {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ (t : Set α), t ⊆ f t)
Goal: ⋃₀ s ⊆ ⋃₀ (f '' s)
-- Set.sUnion_mono_supsets
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
    ⋃₀ (f '' s) ⊆ ⋃₀ s 
Arguments: {α : Type u_1} {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ (t : Set α), f t ⊆ t)
Goal: ⋃₀ (f '' s) ⊆ ⋃₀ s
-- Set.subset_sInter
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S 
Arguments: {α : Type u_1} {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t')
Goal: t ⊆ ⋂₀ S
-- Set.subset_sInter_iff
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' 
Arguments: {α : Type u_1} {S : Set (Set α)} {t : Set α}
Goal: t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t'
-- Set.sUnion_subset_sUnion
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T 
Arguments: {α : Type u_1} {S T : Set (Set α)} (h : S ⊆ T)
Goal: ⋃₀ S ⊆ ⋃₀ T
-- Set.sInter_subset_sInter
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S 
Arguments: {α : Type u_1} {S T : Set (Set α)} (h : S ⊆ T)
Goal: ⋂₀ T ⊆ ⋂₀ S
-- Set.sUnion_empty
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) 
Arguments: {α : Type u_1}
Goal: ⋃₀ ∅ = ∅
-- Set.sInter_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) 
Arguments: {α : Type u_1}
Goal: ⋂₀ ∅ = Set.univ
-- Set.sUnion_singleton
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s 
Arguments: {α : Type u_1} (s : Set α)
Goal: ⋃₀ {s} = s
-- Set.sInter_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s 
Arguments: {α : Type u_1} (s : Set α)
Goal: ⋂₀ {s} = s
-- Set.sUnion_eq_empty
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ 
Arguments: {α : Type u_1} {S : Set (Set α)}
Goal: ⋃₀ S = ∅ ↔ ∀ s ∈ S, s = ∅
-- Set.sInter_eq_univ
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ 
Arguments: {α : Type u_1} {S : Set (Set α)}
Goal: ⋂₀ S = Set.univ ↔ ∀ s ∈ S, s = Set.univ
-- Set.subset_powerset_iff
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t 
Arguments: {α : Type u_1} {s : Set (Set α)} {t : Set α}
Goal: s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t
-- Set.sUnion_powerset_gc
theorem sUnion_powerset_gc :
    GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) 
Arguments: {α : Type u_1}
Goal: GaloisConnection (fun x => ⋃₀ x) fun x => 𝒫 x
-- Set.sUnion_mem_empty_univ
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
    ⋃₀ S ∈ ({∅, univ} : Set (Set α)) 
Arguments: {α : Type u_1} {S : Set (Set α)} (h : S ⊆ {∅, Set.univ})
Goal: ⋃₀ S ∈ {∅, Set.univ}
-- Set.nonempty_sUnion
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s 
Arguments: {α : Type u_1} {S : Set (Set α)}
Goal: (⋃₀ S).Nonempty ↔ ∃ s ∈ S, s.Nonempty
-- Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty 
Arguments: {α : Type u_1} {s : Set (Set α)} (h : (⋃₀ s).Nonempty)
Goal: s.Nonempty
-- Set.Nonempty.of_sUnion_eq_univ
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty 
Arguments: {α : Type u_1} [Nonempty α] {s : Set (Set α)} (h : ⋃₀ s = Set.univ)
Goal: s.Nonempty
-- Set.sUnion_union
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T 
Arguments: {α : Type u_1} (S T : Set (Set α))
Goal: ⋃₀ (S ∪ T) = ⋃₀ S ∪ ⋃₀ T
-- Set.sInter_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T 
Arguments: {α : Type u_1} (S T : Set (Set α))
Goal: ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T
-- Set.sUnion_insert
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T 
Arguments: {α : Type u_1} (s : Set α) (T : Set (Set α))
Goal: ⋃₀ insert s T = s ∪ ⋃₀ T
-- Set.sInter_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T 
Arguments: {α : Type u_1} (s : Set α) (T : Set (Set α))
Goal: ⋂₀ insert s T = s ∩ ⋂₀ T
-- Set.sUnion_diff_singleton_empty
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s 
Arguments: {α : Type u_1} (s : Set (Set α))
Goal: ⋃₀ (s \ {∅}) = ⋃₀ s
-- Set.sInter_diff_singleton_univ
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s 
Arguments: {α : Type u_1} (s : Set (Set α))
Goal: ⋂₀ (s \ {Set.univ}) = ⋂₀ s
-- Set.sUnion_pair
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t 
Arguments: {α : Type u_1} (s t : Set α)
Goal: ⋃₀ {s, t} = s ∪ t
-- Set.sInter_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t 
Arguments: {α : Type u_1} (s t : Set α)
Goal: ⋂₀ {s, t} = s ∩ t
-- Set.sUnion_image
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → Set β) (s : Set α)
Goal: ⋃₀ (f '' s) = ⋃ x ∈ s, f x
-- Set.sInter_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → Set β) (s : Set α)
Goal: ⋂₀ (f '' s) = ⋂ x ∈ s, f x
-- Set.sUnion_range
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x 
Arguments: {β : Type u_2} {ι : Sort u_4} (f : ι → Set β)
Goal: ⋃₀ Set.range f = ⋃ x, f x
-- Set.sInter_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x 
Arguments: {β : Type u_2} {ι : Sort u_4} (f : ι → Set β)
Goal: ⋂₀ Set.range f = ⋂ x, f x
-- Set.iUnion_eq_univ_iff
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i 
Arguments: {α : Type u_1} {ι : Sort u_4} {f : ι → Set α}
Goal: ⋃ i, f i = Set.univ ↔ ∀ (x : α), ∃ i, x ∈ f i
-- Set.iUnion₂_eq_univ_iff
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
    ⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s : (i : ι) → κ i → Set α}
Goal: ⋃ i, ⋃ j, s i j = Set.univ ↔ ∀ (a : α), ∃ i j, a ∈ s i j
-- Set.sUnion_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b 
Arguments: {α : Type u_1} {c : Set (Set α)}
Goal: ⋃₀ c = Set.univ ↔ ∀ (a : α), ∃ b ∈ c, a ∈ b
-- Set.iInter_eq_empty_iff
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i 
Arguments: {α : Type u_1} {ι : Sort u_4} {f : ι → Set α}
Goal: ⋂ i, f i = ∅ ↔ ∀ (x : α), ∃ i, x ∉ f i
-- Set.iInter₂_eq_empty_iff
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
    ⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s : (i : ι) → κ i → Set α}
Goal: ⋂ i, ⋂ j, s i j = ∅ ↔ ∀ (a : α), ∃ i j, a ∉ s i j
-- Set.sInter_eq_empty_iff
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b 
Arguments: {α : Type u_1} {c : Set (Set α)}
Goal: ⋂₀ c = ∅ ↔ ∀ (a : α), ∃ b ∈ c, a ∉ b
-- Set.nonempty_iInter
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i 
Arguments: {α : Type u_1} {ι : Sort u_4} {f : ι → Set α}
Goal: (⋂ i, f i).Nonempty ↔ ∃ x, ∀ (i : ι), x ∈ f i
-- Set.nonempty_iInter₂
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
    (⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s : (i : ι) → κ i → Set α}
Goal: (⋂ i, ⋂ j, s i j).Nonempty ↔ ∃ a, ∀ (i : ι) (j : κ i), a ∈ s i j
-- Set.nonempty_sInter
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b 
Arguments: {α : Type u_1} {c : Set (Set α)}
Goal: (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b
-- Set.compl_sUnion
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) 
Arguments: {α : Type u_1} (S : Set (Set α))
Goal: (⋃₀ S)ᶜ = ⋂₀ (compl '' S)
-- Set.sUnion_eq_compl_sInter_compl
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ 
Arguments: {α : Type u_1} (S : Set (Set α))
Goal: ⋃₀ S = (⋂₀ (compl '' S))ᶜ
-- Set.compl_sInter
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) 
Arguments: {α : Type u_1} (S : Set (Set α))
Goal: (⋂₀ S)ᶜ = ⋃₀ (compl '' S)
-- Set.sInter_eq_compl_sUnion_compl
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ 
Arguments: {α : Type u_1} (S : Set (Set α))
Goal: ⋂₀ S = (⋃₀ (compl '' S))ᶜ
-- Set.inter_empty_of_inter_sUnion_empty
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
    (h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ 
Arguments: {α : Type u_1} {s t : Set α} {S : Set (Set α)} (hs : t ∈ S) (h : s ∩ ⋃₀ S = ∅)
Goal: s ∩ t = ∅
-- Set.range_sigma_eq_iUnion_range
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
    range f = ⋃ a, range fun b => f ⟨a, b⟩ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : α → Type u_11} (f : Sigma γ → β)
Goal: Set.range f = ⋃ a, Set.range fun b => f ⟨a, b⟩
-- Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 
Arguments: {α : Type u_1} {β : Type u_2} (s : α → Set β)
Goal: ⋃ i, s i = Set.range fun a => ↑a.snd
-- Set.iUnion_eq_range_psigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : ι → Set β)
Goal: ⋃ i, s i = Set.range fun a => ↑a.snd
-- Set.iUnion_image_preimage_sigma_mk_eq_self
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
    ⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s 
Arguments: {ι : Type u_11} {σ : ι → Type u_12} (s : Set (Sigma σ))
Goal: ⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s
-- Set.Sigma.univ
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) 
Arguments: {α : Type u_1} (X : α → Type u_11)
Goal: Set.univ = ⋃ a, Set.range (Sigma.mk a)
-- Set.iUnion_subset_iUnion_const
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι₂ : Sort u_6} {s : Set α} (h : ι → ι₂)
Goal: ⋃ x, s ⊆ ⋃ x, s
-- Set.iUnion_singleton_eq_range
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f 
Arguments: {α : Type u_11} {β : Type u_12} (f : α → β)
Goal: ⋃ x, {f x} = Set.range f
-- Set.iUnion_of_singleton
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ 
Arguments: (α : Type u_11)
Goal: ⋃ x, {x} = Set.univ
-- Set.iUnion_of_singleton_coe
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s 
Arguments: {α : Type u_1} (s : Set α)
Goal: ⋃ i, {↑i} = s
-- Set.sUnion_eq_biUnion
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i 
Arguments: {α : Type u_1} {s : Set (Set α)}
Goal: ⋃₀ s = ⋃ i ∈ s, i
-- Set.sInter_eq_biInter
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i 
Arguments: {α : Type u_1} {s : Set (Set α)}
Goal: ⋂₀ s = ⋂ i ∈ s, i
-- Set.sUnion_eq_iUnion
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i 
Arguments: {α : Type u_1} {s : Set (Set α)}
Goal: ⋃₀ s = ⋃ i, ↑i
-- Set.sInter_eq_iInter
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i 
Arguments: {α : Type u_1} {s : Set (Set α)}
Goal: ⋂₀ s = ⋂ i, ↑i
-- Set.iUnion_of_empty
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ 
Arguments: {α : Type u_1} {ι : Sort u_4} [IsEmpty ι] (s : ι → Set α)
Goal: ⋃ i, s i = ∅
-- Set.iInter_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ 
Arguments: {α : Type u_1} {ι : Sort u_4} [IsEmpty ι] (s : ι → Set α)
Goal: ⋂ i, s i = Set.univ
-- Set.union_eq_iUnion
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ 
Arguments: {α : Type u_1} {s₁ s₂ : Set α}
Goal: s₁ ∪ s₂ = ⋃ b, bif b then s₁ else s₂
-- Set.inter_eq_iInter
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ 
Arguments: {α : Type u_1} {s₁ s₂ : Set α}
Goal: s₁ ∩ s₂ = ⋂ b, bif b then s₁ else s₂
-- Set.sInter_union_sInter
theorem sInter_union_sInter {S T : Set (Set α)} :
    ⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 
Arguments: {α : Type u_1} {S T : Set (Set α)}
Goal: ⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, p.1 ∪ p.2
-- Set.sUnion_inter_sUnion
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
    ⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 
Arguments: {α : Type u_1} {s t : Set (Set α)}
Goal: ⋃₀ s ∩ ⋃₀ t = ⋃ p ∈ s ×ˢ t, p.1 ∩ p.2