awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
refine_1 :
  ∀ (u : Set (Compacts α)),
    Membership.mem (γ := Set (Set (Compacts α)))
        ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
          {u : Set (Set α) | u.Finite ∧ u ⊆ B})
        u →
      IsOpen u
refine_2 :
  ∀ (u : Set (Compacts α)),
    Membership.mem (γ := Set (Set (Compacts α)))
        (preimage SetLike.coe ''
          (fun (p : Set α × Set (Set α)) ↦ {s : Set α | s ⊆ p.1 ∧ ∀ U ∈ p.2, (s ∩ U).Nonempty}) ''
            {p : Set α × Set (Set α) | IsOpen[inst] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1})
        u →
      ∀ x ∈ u,
        ∃ (v : Set (Compacts α)),
          Membership.mem (γ := Set (Set (Compacts α)))
              ((fun (u : Set (Set α)) ↦
                  {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
                {u : Set (Set α) | u.Finite ∧ u ⊆ B})
              v ∧
            x ∈ v ∧ v ⊆ u
⊢ IsTopologicalBasis.{u_1} (α := Compacts α)
    ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
      {u : Set (Set α) | u.Finite ∧ u ⊆ B})

Tactic:
refine hB.vietoris.isInducing isEmbedding_coe.isInducing
    |>.isTopologicalBasis_of_exists_subset ?_ ?_

Premises used: TopologicalSpace.IsTopologicalBasis.vietoris, TopologicalSpace.Compacts.isEmbedding_coe, Topology.IsEmbedding.isInducing, TopologicalSpace.IsTopologicalBasis.isInducing, TopologicalSpace.IsTopologicalBasis.isTopologicalBasis_of_exists_subset
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
refine_1 :
  ∀ ⦃x : Set (Set α)⦄,
    Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ u ⊆ B} x →
      IsOpen.{u_1} (X := Compacts α) {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ x ∧ ∀ U ∈ x, ((↑K : Set α) ∩ U).Nonempty}
⊢ ∀ (u : Set (Compacts α)),
    Membership.mem (γ := Set (Set (Compacts α)))
        ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
          {u : Set (Set α) | u.Finite ∧ u ⊆ B})
        u →
      IsOpen u

Tactic:
simp_rw [forall_mem_image]

Premises used: Set.forall_mem_image
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
refine_2 :
  ∀ ⦃x : Set α × Set (Set α)⦄,
    Membership.mem (γ := Set (Set α × Set (Set α)))
        {p : Set α × Set (Set α) | IsOpen[inst] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1} x →
      ∀ (x_1 : Compacts α),
        Membership.mem (γ := Set (Compacts α)) (SetLike.coe ⁻¹' {s : Set α | s ⊆ x.1 ∧ ∀ U ∈ x.2, (s ∩ U).Nonempty})
            x_1 →
          ∃ (v : Set (Compacts α)),
            Membership.mem (γ := Set (Set (Compacts α)))
                ((fun (u : Set (Set α)) ↦
                    {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
                  {u : Set (Set α) | u.Finite ∧ u ⊆ B})
                v ∧
              x_1 ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ x.1 ∧ ∀ U ∈ x.2, (s ∩ U).Nonempty}
⊢ ∀ (u : Set (Compacts α)),
    Membership.mem (γ := Set (Set (Compacts α)))
        (preimage SetLike.coe ''
          (fun (p : Set α × Set (Set α)) ↦ {s : Set α | s ⊆ p.1 ∧ ∀ U ∈ p.2, (s ∩ U).Nonempty}) ''
            {p : Set α × Set (Set α) | IsOpen[inst] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1})
        u →
      ∀ x ∈ u,
        ∃ (v : Set (Compacts α)),
          Membership.mem (γ := Set (Set (Compacts α)))
              ((fun (u : Set (Set α)) ↦
                  {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
                {u : Set (Set α) | u.Finite ∧ u ⊆ B})
              v ∧
            x ∈ v ∧ v ⊆ u

Tactic:
simp_rw [forall_mem_image]

Premises used: Set.forall_mem_image
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
refine_1 :
  ∀ (hu : u.Finite) (huB : u ⊆ B),
    IsOpen.{u_1} (X := Compacts α) {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}
⊢ Membership.mem (γ := Set (Set (Set α))) {u : Set (Set α) | u.Finite ∧ u ⊆ B} u →
    IsOpen.{u_1} (X := Compacts α) {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}

Tactic:
⟨hu, huB⟩
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu : u.Finite
huB : u ⊆ B
refine_1 :
  IsOpen.{u_1} (X := Compacts α)
    ({a : Compacts α | (↑a : Set α) ⊆ ⋃₀ u} ∩ ⋂ i ∈ u, {x : Compacts α | ((↑x : Set α) ∩ i).Nonempty})
⊢ IsOpen.{u_1} (X := Compacts α) {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}

Tactic:
simp_rw [setOf_and, setOf_forall]

Premises used: Set.setOf_and, Set.setOf_forall
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu : u.Finite
huB : u ⊆ B
⊢ IsOpen.{u_1} (X := Compacts α)
    ({a : Compacts α | (↑a : Set α) ⊆ ⋃₀ u} ∩ ⋂ i ∈ u, {x : Compacts α | ((↑x : Set α) ∩ i).Nonempty})

Tactic:
.inter
      (isOpen_subsets_of_isOpen <| isOpen_sUnion fun U hU => hB.isOpen <| huB hU)
      (hu.isOpen_biInter fun U hU => isOpen_inter_nonempty_of_isOpen <| hB.isOpen <| huB hU)

Premises used: TopologicalSpace.IsTopologicalBasis.isOpen, TopologicalSpace.Compacts.isOpen_inter_nonempty_of_isOpen, IsOpen.inter, TopologicalSpace.Compacts.isOpen_subsets_of_isOpen, Set.Finite.isOpen_biInter, isOpen_sUnion
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu : u.Finite
huB : u ⊆ B
⊢ IsOpen.{u_1} (X := Compacts α)
    ({a : Compacts α | (↑a : Set α) ⊆ ⋃₀ u} ∩ ⋂ i ∈ u, {x : Compacts α | ((↑x : Set α) ∩ i).Nonempty})

Tactic:
exact .inter
      (isOpen_subsets_of_isOpen <| isOpen_sUnion fun U hU => hB.isOpen <| huB hU)
      (hu.isOpen_biInter fun U hU => isOpen_inter_nonempty_of_isOpen <| hB.isOpen <| huB hU)
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
refine_2 :
  ∀ (hV : IsOpen[inst] (V, u).1) (hu : (V, u).2.Finite) (huB : (V, u).2 ⊆ B) (huV : ∀ U ∈ (V, u).2, U ⊆ (V, u).1)
    (x : Compacts α),
    Membership.mem (γ := Set (Compacts α))
        (SetLike.coe ⁻¹' {s : Set α | s ⊆ (V, u).1 ∧ ∀ U ∈ (V, u).2, (s ∩ U).Nonempty}) x →
      ∃ (v : Set (Compacts α)),
        Membership.mem (γ := Set (Set (Compacts α)))
            ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
              {u : Set (Set α) | u.Finite ∧ u ⊆ B})
            v ∧
          x ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ (V, u).1 ∧ ∀ U ∈ (V, u).2, (s ∩ U).Nonempty}
⊢ Membership.mem (γ := Set (Set α × Set (Set α)))
      {p : Set α × Set (Set α) | IsOpen[inst] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1} (V, u) →
    ∀ (x : Compacts α),
      Membership.mem (γ := Set (Compacts α))
          (SetLike.coe ⁻¹' {s : Set α | s ⊆ (V, u).1 ∧ ∀ U ∈ (V, u).2, (s ∩ U).Nonempty}) x →
        ∃ (v : Set (Compacts α)),
          Membership.mem (γ := Set (Set (Compacts α)))
              ((fun (u : Set (Set α)) ↦
                  {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
                {u : Set (Set α) | u.Finite ∧ u ⊆ B})
              v ∧
            x ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ (V, u).1 ∧ ∀ U ∈ (V, u).2, (s ∩ U).Nonempty}

Tactic:
⟨hV, hu, huB, huV⟩
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] (V, u).1
hu : (V, u).2.Finite
huB : (V, u).2 ⊆ B
huV : ∀ U ∈ (V, u).2, U ⊆ (V, u).1
K : Compacts α
refine_2 :
  ∀ (hKV : (↑K : Set α) ⊆ (V, u).1) (hKu : ∀ U ∈ (V, u).2, ((↑K : Set α) ∩ U).Nonempty),
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ (V, u).1 ∧ ∀ U ∈ (V, u).2, (s ∩ U).Nonempty}
⊢ Membership.mem (γ := Set (Compacts α)) (SetLike.coe ⁻¹' {s : Set α | s ⊆ (V, u).1 ∧ ∀ U ∈ (V, u).2, (s ∩ U).Nonempty})
      K →
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ (V, u).1 ∧ ∀ U ∈ (V, u).2, (s ∩ U).Nonempty}

Tactic:
⟨hKV, hKu⟩
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] (V, u).1
hu : (V, u).2.Finite
huB : (V, u).2 ⊆ B
huV : ∀ U ∈ (V, u).2, U ⊆ (V, u).1
K : Compacts α
hKV : (↑K : Set α) ⊆ (V, u).1
hKu : ∀ U ∈ (V, u).2, ((↑K : Set α) ∩ U).Nonempty
refine_2 :
  ∃ (v : Set (Compacts α)),
    Membership.mem (γ := Set (Set (Compacts α)))
        ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
          {u : Set (Set α) | u.Finite ∧ u ⊆ B})
        v ∧
      K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
⊢ ∃ (v : Set (Compacts α)),
    Membership.mem (γ := Set (Set (Compacts α)))
        ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
          {u : Set (Set α) | u.Finite ∧ u ⊆ B})
        v ∧
      K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ (V, u).1 ∧ ∀ U ∈ (V, u).2, (s ∩ U).Nonempty}

Tactic:
dsimp only at *
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKV : (↑K : Set α) ⊆ V
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
refine_2 :
  ∀ (w : Set (Set α)) (hwB : w ⊆ B) (hwV : V = ⋃₀ w),
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
⊢ ∃ (v : Set (Compacts α)),
    Membership.mem (γ := Set (Set (Compacts α)))
        ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
          {u : Set (Set α) | u.Finite ∧ u ⊆ B})
        v ∧
      K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
obtain ⟨w, hwB, hwV⟩ := hB.open_eq_sUnion hV

Premises used: TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKV : (↑K : Set α) ⊆ V
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hwB : w ⊆ B
hwV : V = ⋃₀ w
⊢ (↑K : Set α) ⊆ ⋃₀ w

Tactic:
rw [hwV] at hKV
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKV : (↑K : Set α) ⊆ V
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hwB : w ⊆ B
hwV : V = ⋃₀ w
hKV₁ : (↑K : Set α) ⊆ ⋃₀ w
⊢ w ⊆ B

Tactic:
rw [hwV] at hKV
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKV : (↑K : Set α) ⊆ V
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hwB : w ⊆ B
hwV : V = ⋃₀ w
hKV₁ : (↑K : Set α) ⊆ ⋃₀ w
hwB₁ : w ⊆ B
⊢ V = ⋃₀ w

Tactic:
rw [hwV] at hKV
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hKV : (↑K : Set α) ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : V = ⋃₀ w
⊢ ⋃₀ w ⊆ V

Tactic:
replace hwV := hwV.symm.subset
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hKV : (↑K : Set α) ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
refine_2 :
  ∀
    (this :
      ∀ (w : Set (Set α)) (hKV : (↑K : Set α) ⊆ ⋃₀ w) (hwB : w ⊆ B) (hwV : ⋃₀ w ⊆ V)
        (hwK : ∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty),
        ∃ (v : Set (Compacts α)),
          Membership.mem (γ := Set (Set (Compacts α)))
              ((fun (u : Set (Set α)) ↦
                  {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
                {u : Set (Set α) | u.Finite ∧ u ⊆ B})
              v ∧
            K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty})
    (hwK : ¬∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty),
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
wlog :
  ∀ (w : Set (Set α)) (hKV : (↑K : Set α) ⊆ ⋃₀ w) (hwB : w ⊆ B) (hwV : ⋃₀ w ⊆ V)
    (hwK : ∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty),
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
⊢ ∃ (v : Set (Compacts α)),
    Membership.mem (γ := Set (Set (Compacts α)))
        ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
          {u : Set (Set α) | u.Finite ∧ u ⊆ B})
        v ∧
      K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
wlog hwK : ∀ W ∈ w, ((K : Set α) ∩ W).Nonempty generalizing w
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hKV : (↑K : Set α) ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)) (hKV : (↑K : Set α) ⊆ ⋃₀ w) (hwB : w ⊆ B) (hwV : ⋃₀ w ⊆ V)
    (hwK : ∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty),
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty
⊢ {W : Set α | W ∈ w ∧ ((↑K : Set α) ∩ W).Nonempty} ⊆ B

Tactic:
grind
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hKV : (↑K : Set α) ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)) (hKV : (↑K : Set α) ⊆ ⋃₀ w) (hwB : w ⊆ B) (hwV : ⋃₀ w ⊆ V)
    (hwK : ∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty),
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty
⊢ ⋃₀ {W : Set α | W ∈ w ∧ ((↑K : Set α) ∩ W).Nonempty} ⊆ V

Tactic:
grind
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hKV : (↑K : Set α) ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)) (hKV : (↑K : Set α) ⊆ ⋃₀ w) (hwB : w ⊆ B) (hwV : ⋃₀ w ⊆ V)
    (hwK : ∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty),
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty
⊢ ∀ (W : Set α),
    Membership.mem (γ := Set (Set α)) {W : Set α | W ∈ w ∧ ((↑K : Set α) ∩ W).Nonempty} W → ((↑K : Set α) ∩ W).Nonempty

Tactic:
grind
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hKV : (↑K : Set α) ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)) (hKV : (↑K : Set α) ⊆ ⋃₀ w) (hwB : w ⊆ B) (hwV : ⋃₀ w ⊆ V)
    (hwK : ∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty),
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty
refine_2 :
  ∀ (x : α) (hx : x ∈ (↑K : Set α)),
    Membership.mem (γ := Set α) (⋃₀ {W : Set α | W ∈ w ∧ ((↑K : Set α) ∩ W).Nonempty}) x
⊢ ∃ (v : Set (Compacts α)),
    Membership.mem (γ := Set (Set (Compacts α)))
        ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
          {u : Set (Set α) | u.Finite ∧ u ⊆ B})
        v ∧
      K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
refine this {W ∈ w | (↑K ∩ W).Nonempty} (fun x hx => ?_) (by grind) (by grind) (by grind)

Premises used: setOf, Membership.mem, Inter.inter, And, Set.Nonempty
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hKV : (↑K : Set α) ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)) (hKV : (↑K : Set α) ⊆ ⋃₀ w) (hwB : w ⊆ B) (hwV : ⋃₀ w ⊆ V)
    (hwK : ∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty),
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty
x : α
hx : x ∈ (↑K : Set α)
refine_2 :
  ∀ (W : Set α) (hWw : W ∈ w) (hxW : x ∈ W),
    Membership.mem (γ := Set α) (⋃₀ {W : Set α | W ∈ w ∧ ((↑K : Set α) ∩ W).Nonempty}) x
⊢ Membership.mem (γ := Set α) (⋃₀ {W : Set α | W ∈ w ∧ ((↑K : Set α) ∩ W).Nonempty}) x

Tactic:
obtain ⟨W, hWw, hxW⟩ := mem_sUnion.mp (hKV hx)

Premises used: Iff.mp, Set.mem_sUnion
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hKV : (↑K : Set α) ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)) (hKV : (↑K : Set α) ⊆ ⋃₀ w) (hwB : w ⊆ B) (hwV : ⋃₀ w ⊆ V)
    (hwK : ∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty),
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty
x : α
hx : x ∈ (↑K : Set α)
W : Set α
hWw : W ∈ w
hxW : x ∈ W
⊢ Membership.mem (γ := Set α) (⋃₀ {W : Set α | W ∈ w ∧ ((↑K : Set α) ∩ W).Nonempty}) x

Tactic:
mem_sUnion_of_mem hxW ⟨hWw, x, hx, hxW⟩

Premises used: Set.mem_sUnion_of_mem, And.intro, Exists.intro
-- Module: Mathlib.Topology.Sets.VietorisTopology
Goal:
α : Type u_1
inst : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty
w : Set (Set α)
hKV : (↑K : Set α) ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)) (hKV : (↑K : Set α) ⊆ ⋃₀ w) (hwB : w ⊆ B) (hwV : ⋃₀ w ⊆ V)
    (hwK : ∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty),
    ∃ (v : Set (Compacts α)),
      Membership.mem (γ := Set (Set (Compacts α)))
          ((fun (u : Set (Set α)) ↦ {K : Compacts α | (↑K : Set α) ⊆ ⋃₀ u ∧ ∀ U ∈ u, ((↑K : Set α) ∩ U).Nonempty}) ''
            {u : Set (Set α) | u.Finite ∧ u ⊆ B})
          v ∧
        K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s : Set α | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, ((↑K : Set α) ∩ W).Nonempty
x : α
hx : x ∈ (↑K : Set α)
W : Set α
hWw : W ∈ w
hxW : x ∈ W
⊢ Membership.mem (γ := Set α) (⋃₀ {W : Set α | W ∈ w ∧ ((↑K : Set α) ∩ W).Nonempty}) x

Tactic:
exact mem_sUnion_of_mem hxW ⟨hWw, x, hx, hxW⟩