awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ :
  t ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
s : Set α
hs : s ∈ ⋂₀ t
refine_2 :
  ∃ (x : Set (Set α)),
    Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
      Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
        {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ t
⊢ ∃ (v : Set (Set α)),
    Membership.mem (γ := Set (Set (Set α)))
        ((fun (u : Set (Set α)) ↦ {s : Set α | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}) ''
          {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U})
        v ∧
      s ∈ v ∧ v ⊆ ⋂₀ t

Tactic:
rw [exists_mem_image]

Premises used: Set.exists_mem_image
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ :
  t ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
s : Set α
hs : s ∈ ⋂₀ t
refine_2 :
  ∀ (hs : ∅ ∈ ⋂₀ t),
    ∃ (x : Set (Set α)),
      Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
        Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ∅ ∧
          {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ t
refine_2₁ :
  s.Nonempty →
    ∃ (x : Set (Set α)),
      Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
        Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
          {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ t
⊢ ∃ (x : Set (Set α)),
    Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
      Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
        {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ t

Tactic:
rcases eq_empty_or_nonempty s with rfl | _

Premises used: Set.eq_empty_or_nonempty, rfl
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ :
  t ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
hs : ∅ ∈ ⋂₀ t
refine_2 :
  Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} ∅ ∧
    Membership.mem (γ := Set (Set α))
        {s : Set α | s ⊆ ⋃₀ ∅ ∧ ∀ (U : Set α), Membership.mem (γ := Set (Set α)) ∅ U → (s ∩ U).Nonempty} ∅ ∧
      {s : Set α | s ⊆ ⋃₀ ∅ ∧ ∀ (U : Set α), Membership.mem (γ := Set (Set α)) ∅ U → (s ∩ U).Nonempty} ⊆ ⋂₀ t
⊢ ∃ (x : Set (Set α)),
    Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
      Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ∅ ∧
        {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ t

Tactic:
exists ∅
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ :
  t ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
hs : ∅ ∈ ⋂₀ t
⊢ Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} ∅ ∧
    Membership.mem (γ := Set (Set α))
        {s : Set α | s ⊆ ⋃₀ ∅ ∧ ∀ (U : Set α), Membership.mem (γ := Set (Set α)) ∅ U → (s ∩ U).Nonempty} ∅ ∧
      {s : Set α | s ⊆ ⋃₀ ∅ ∧ ∀ (U : Set α), Membership.mem (γ := Set (Set α)) ∅ U → (s ∩ U).Nonempty} ⊆ ⋂₀ t

Tactic:
simpa
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ :
  t ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
s : Set α
hs : s ∈ ⋂₀ t
h : s.Nonempty
rw :
  ∀
    (ht₂ :
      t ∩ powerset '' {U : Set α | IsOpen[inst] U} ∪
          t ∩ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V} =
        t)
    (s : Set α) (hs : s ∈ ⋂₀ t),
    s.Nonempty →
      ∃ (u : Set (Set (Set α))) (v : Set (Set (Set α))),
        u ⊆ powerset '' {U : Set α | IsOpen[inst] U} ∧
          v ⊆ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U} ∧ u ∪ v = t
⊢ ∃ (u : Set (Set (Set α))) (v : Set (Set (Set α))),
    u ⊆ powerset '' {U : Set α | IsOpen[inst] U} ∧
      v ⊆ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U} ∧ u ∪ v = t

Tactic:
rw [← inter_eq_left, inter_union_distrib_left] at ht₂

Premises used: Set.inter_eq_left, Set.inter_union_distrib_left
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ :
  t ∩ powerset '' {U : Set α | IsOpen[inst] U} ∪
      t ∩ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V} =
    t
s : Set α
hs : s ∈ ⋂₀ t
h : s.Nonempty
⊢ ∃ (u : Set (Set (Set α))) (v : Set (Set (Set α))),
    u ⊆ powerset '' {U : Set α | IsOpen[inst] U} ∧
      v ⊆ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U} ∧ u ∪ v = t

Tactic:
⟨_, _, inter_subset_right, inter_subset_right, ht₂⟩

Premises used: And.intro, Exists.intro, Set.inter_subset_right
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ :
  t ∩ powerset '' {U : Set α | IsOpen[inst] U} ∪
      t ∩ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V} =
    t
s : Set α
hs : s ∈ ⋂₀ t
h : s.Nonempty
⊢ ∃ (u : Set (Set (Set α))) (v : Set (Set (Set α))),
    u ⊆ powerset '' {U : Set α | IsOpen[inst] U} ∧
      v ⊆ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U} ∧ u ∪ v = t

Tactic:
exact ⟨_, _, inter_subset_right, inter_subset_right, ht₂⟩
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ :
  t ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
s : Set α
hs : s ∈ ⋂₀ t
h : s.Nonempty
refine_2 :
  ∀ (u v : Set (Set (Set α))) (hu : u ⊆ powerset '' {U : Set α | IsOpen[inst] U})
    (hv : v ⊆ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U}) (ht₁ : (u ∪ v).Finite)
    (ht₂ :
      u ∪ v ⊆
        powerset '' {U : Set α | IsOpen[inst] U} ∪
          (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V})
    (hs : s ∈ ⋂₀ (u ∪ v)),
    ∃ (x : Set (Set α)),
      Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
        Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
          {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ (u ∪ v)
⊢ ∃ (x : Set (Set α)),
    Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
      Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
        {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ t

Tactic:
obtain ⟨u, v, hu, hv, rfl⟩ : ∃ u v, u ⊆ powerset '' {U | IsOpen U} ∧
        v ⊆ (fun V => {s | (s ∩ V).Nonempty}) '' {U | IsOpen U} ∧ u ∪ v = t := by
      rw [← inter_eq_left, inter_union_distrib_left] at ht₂
      exact ⟨_, _, inter_subset_right, inter_subset_right, ht₂⟩

Premises used: setOf, Set.powerset, rfl
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h : s.Nonempty
u v : Set (Set (Set α))
hu : u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
hv : v ⊆ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U}
ht₁ : (u ∪ v).Finite
ht₂ :
  u ∪ v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
hs : s ∈ ⋂₀ (u ∪ v)
⊢ u.Finite ∧ v.Finite

Tactic:
rw [finite_union] at ht₁

Premises used: Set.finite_union
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h : s.Nonempty
u v : Set (Set (Set α))
hu : u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
hv : v ⊆ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U}
ht₁ : (u ∪ v).Finite
ht₂ :
  u ∪ v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
hs : s ∈ ⋂₀ (u ∪ v)
ht₃ : u.Finite ∧ v.Finite
⊢ u ∪ v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}

Tactic:
rw [finite_union] at ht₁

Premises used: Set.finite_union
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h : s.Nonempty
u v : Set (Set (Set α))
hu : u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
hv : v ⊆ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U}
ht₁ : (u ∪ v).Finite
ht₂ :
  u ∪ v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
hs : s ∈ ⋂₀ (u ∪ v)
ht₃ : u.Finite ∧ v.Finite
ht₄ :
  u ∪ v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
⊢ s ∈ ⋂₀ (u ∪ v)

Tactic:
rw [finite_union] at ht₁

Premises used: Set.finite_union
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h : s.Nonempty
u v : Set (Set (Set α))
hu : u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
hv : v ⊆ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U}
ht₁ : u.Finite ∧ v.Finite
ht₂ :
  u ∪ v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
hs : s ∈ ⋂₀ (u ∪ v)
refine_2 :
  ∀ (u : Set (Set α)) (hu₁ : u ⊆ {U : Set α | IsOpen[inst] U}) (hu₂ : u.Finite)
    (hu : powerset '' u ⊆ powerset '' {U : Set α | IsOpen[inst] U}) (ht₁ : (powerset '' u).Finite ∧ v.Finite)
    (ht₂ :
      powerset '' u ∪ v ⊆
        powerset '' {U : Set α | IsOpen[inst] U} ∪
          (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V})
    (hs : s ∈ ⋂₀ (powerset '' u ∪ v)),
    ∃ (x : Set (Set α)),
      Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
        Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
          {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ (powerset '' u ∪ v)
⊢ ∃ (x : Set (Set α)),
    Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
      Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
        {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ (u ∪ v)

Tactic:
obtain ⟨u, hu₁, hu₂, rfl⟩ := exists_subset_image_finite_and.mp ⟨u, hu, ht₁.1, rfl⟩

Premises used: And.left, And.intro, Exists.intro, Iff.mp, Set.exists_subset_image_finite_and, rfl
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h : s.Nonempty
v : Set (Set (Set α))
hv : v ⊆ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U}
u : Set (Set α)
hu₁ : u ⊆ {U : Set α | IsOpen[inst] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
ht₁ : (powerset '' u).Finite ∧ v.Finite
ht₂ :
  powerset '' u ∪ v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
hs : s ∈ ⋂₀ (powerset '' u ∪ v)
refine_2 :
  ∀ (v : Set (Set α)) (hv₁ : v ⊆ {U : Set α | IsOpen[inst] U}) (hv₂ : v.Finite)
    (hv :
      Subset (α := Set (Set (Set α))) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
        ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U}))
    (ht₁ :
      (powerset '' u).Finite ∧ Set.Finite (α := Set (Set α)) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v))
    (ht₂ :
      powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v ⊆
        powerset '' {U : Set α | IsOpen[inst] U} ∪
          (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V})
    (hs : s ∈ ⋂₀ (powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)),
    ∃ (x : Set (Set α)),
      Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
        Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
          {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆
            ⋂₀ (powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
⊢ ∃ (x : Set (Set α)),
    Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
      Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
        {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ (powerset '' u ∪ v)

Tactic:
obtain ⟨v, hv₁, hv₂, rfl⟩ := exists_subset_image_finite_and.mp ⟨v, hv, ht₁.2, rfl⟩

Premises used: And.right, And.intro, Exists.intro, Iff.mp, Set.exists_subset_image_finite_and, rfl
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U : Set α | IsOpen[inst] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
v : Set (Set α)
hv₁ : v ⊆ {U : Set α | IsOpen[inst] U}
hv₂ : v.Finite
hv :
  Subset (α := Set (Set (Set α))) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
    ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U})
ht₁ : (powerset '' u).Finite ∧ Set.Finite (α := Set (Set α)) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
ht₂ :
  powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
hs : s ∈ ⋂₀ (powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
U : Set α := ⋂₀ u
⊢ IsOpen[inst] U

Tactic:
have hU : IsOpen U := hu₂.isOpen_sInter hu₁
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h_1 : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U : Set α | IsOpen[inst] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
v : Set (Set α)
hv₁ : v ⊆ {U : Set α | IsOpen[inst] U}
hv₂ : v.Finite
hv :
  Subset (α := Set (Set (Set α))) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
    ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U})
ht₁ : (powerset '' u).Finite ∧ Set.Finite (α := Set (Set α)) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
ht₂ :
  powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst] U
hs : s ∈ ⋂₀ (powerset '' u) ∩ ⋂₀ ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
h : ∀ (x : Set α), x ∈ ⋂₀ (powerset '' u) ↔ x ∈ 𝒫 U
⊢ ⋂₀ (powerset '' u) = 𝒫 U

Tactic:
ext
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U : Set α | IsOpen[inst] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
v : Set (Set α)
hv₁ : v ⊆ {U : Set α | IsOpen[inst] U}
hv₂ : v.Finite
hv :
  Subset (α := Set (Set (Set α))) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
    ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U})
ht₁ : (powerset '' u).Finite ∧ Set.Finite (α := Set (Set α)) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
ht₂ :
  powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst] U
hs : s ∈ ⋂₀ (powerset '' u) ∩ ⋂₀ ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
x : Set α
⊢ x ∈ ⋂₀ (powerset '' u) ↔ x ∈ 𝒫 U

Tactic:
simp [U]
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U : Set α | IsOpen[inst] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
v : Set (Set α)
hv₁ : v ⊆ {U : Set α | IsOpen[inst] U}
hv₂ : v.Finite
hv :
  Subset (α := Set (Set (Set α))) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
    ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U})
ht₁ : (powerset '' u).Finite ∧ Set.Finite (α := Set (Set α)) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
ht₂ :
  powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
hs : s ∈ ⋂₀ (powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
U : Set α := ⋂₀ u
hU : IsOpen[inst] U
refine_2 :
  ∀ (hs : s ∈ 𝒫 U ∩ ⋂ a ∈ v, {s : Set α | (s ∩ a).Nonempty}),
    ∃ (x : Set (Set α)),
      Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
        Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
          {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ 𝒫 U ∩ ⋂ a ∈ v, {s : Set α | (s ∩ a).Nonempty}
⊢ ∃ (x : Set (Set α)),
    Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
      Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
        {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆
          ⋂₀ (powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)

Tactic:
simp_rw [sInter_union, show ⋂₀ (powerset '' u) = U.powerset by ext; simp [U], sInter_image]
      at hs ⊢

Premises used: Set.sInter_union, Set.sInter_image
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U : Set α | IsOpen[inst] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
v : Set (Set α)
hv₁ : v ⊆ {U : Set α | IsOpen[inst] U}
hv₂ : v.Finite
hv :
  Subset (α := Set (Set (Set α))) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
    ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U})
ht₁ : (powerset '' u).Finite ∧ Set.Finite (α := Set (Set α)) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
ht₂ :
  powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst] U
hs : s ∈ 𝒫 U ∩ ⋂ a ∈ v, {s : Set α | (s ∩ a).Nonempty}
⊢ s ⊆ U ∧ ∀ (i : Set α) (j : i ∈ v), Membership.mem (γ := Set (Set α)) {s : Set α | (s ∩ i).Nonempty} s

Tactic:
rw [mem_inter_iff, mem_powerset_iff, mem_iInter₂] at hs

Premises used: Set.mem_powerset_iff, Set.mem_inter_iff, Set.mem_iInter₂
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U : Set α | IsOpen[inst] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
v : Set (Set α)
hv₁ : v ⊆ {U : Set α | IsOpen[inst] U}
hv₂ : v.Finite
hv :
  Subset (α := Set (Set (Set α))) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
    ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U})
ht₁ : (powerset '' u).Finite ∧ Set.Finite (α := Set (Set α)) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
ht₂ :
  powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst] U
hs : s ⊆ U ∧ ∀ (i : Set α) (j : i ∈ v), Membership.mem (γ := Set (Set α)) {s : Set α | (s ∩ i).Nonempty} s
refine_2 :
  Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U}
      (insert U ((fun (x : Set α) ↦ U ∩ x) '' v)) ∧
    Membership.mem (γ := Set (Set α))
        {s : Set α |
          s ⊆ ⋃₀ insert U ((fun (x : Set α) ↦ U ∩ x) '' v) ∧
            ∀ (U_1 : Set α),
              Membership.mem (γ := Set (Set α)) (insert U ((fun (x : Set α) ↦ U ∩ x) '' v)) U_1 → (s ∩ U_1).Nonempty}
        s ∧
      {s : Set α |
          s ⊆ ⋃₀ insert U ((fun (x : Set α) ↦ U ∩ x) '' v) ∧
            ∀ (U_1 : Set α),
              Membership.mem (γ := Set (Set α)) (insert U ((fun (x : Set α) ↦ U ∩ x) '' v)) U_1 → (s ∩ U_1).Nonempty} ⊆
        𝒫 U ∩ ⋂ a ∈ v, {s : Set α | (s ∩ a).Nonempty}
⊢ ∃ (x : Set (Set α)),
    Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ ∀ U ∈ u, IsOpen[inst] U} x ∧
      Membership.mem (γ := Set (Set α)) {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} s ∧
        {s : Set α | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ 𝒫 U ∩ ⋂ a ∈ v, {s : Set α | (s ∩ a).Nonempty}

Tactic:
exists insert U ((U ∩ ·) '' v)

Premises used: Insert.insert, Inter.inter, Set.image
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
s : Set α
h : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U : Set α | IsOpen[inst] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U : Set α | IsOpen[inst] U}
v : Set (Set α)
hv₁ : v ⊆ {U : Set α | IsOpen[inst] U}
hv₂ : v.Finite
hv :
  Subset (α := Set (Set (Set α))) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
    ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {U : Set α | IsOpen[inst] U})
ht₁ : (powerset '' u).Finite ∧ Set.Finite (α := Set (Set α)) ((fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v)
ht₂ :
  powerset '' u ∪ (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U : Set α | IsOpen[inst] U} ∪
      (fun (V : Set α) ↦ {s : Set α | (s ∩ V).Nonempty}) '' {V : Set α | IsOpen[inst] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst] U
hs : s ⊆ U ∧ ∀ (i : Set α) (j : i ∈ v), Membership.mem (γ := Set (Set α)) {s : Set α | (s ∩ i).Nonempty} s
⊢ ⋃₀ insert U ((fun (x : Set α) ↦ U ∩ x) '' v) = U

Tactic:
simp