-- Module: Mathlib.Topology.Sheaves.EtaleSpace
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
U : Opens (↑X : Type v)
hF_bij :
  ∀ (x : (↑X : Type v)) (hx : x ∈ U),
    Bijective.{v + 1, v + 1} (α := CC (F.obj (op U))) (β := CC (F.stalk x))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx)) : (a : CC (F.obj (op U))) → CC (F.stalk x))
x : (↑X : Type v)
hx : x ∈ U
y : ↥U
g : ToType (FC := FC) (CC := CC) (F.stalk x)
V : Opens (↑X : Type v)
f : ToType (FC := FC) (CC := CC) (F.obj (op V))
hyV : (↑y : (↑X : Type v)) ∈ V
hgf :
  Eq (α := CC (F.stalk (↑y : (↑X : Type v))))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (↑y : (↑X : Type v)) ⋯) :
          (a : CC (F.obj (op U))) → CC (F.stalk (↑y : (↑X : Type v))))
        (surjInv ⋯ g) :
      CC (F.stalk (↑y : (↑X : Type v))))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V (↑y : (↑X : Type v)) ⋯) :
          (a : CC (F.obj (op V))) → CC (F.stalk (↑y : (↑X : Type v))))
        f :
      CC (F.stalk (↑y : (↑X : Type v))))
rcases :
  ∀ (W : Opens (↑X : Type v)) (hyW : (↑y : (↑X : Type v)) ∈ W) (ιWU : W ⟶ U) (ιWV : W ⟶ V)
    (hW :
      Eq (α := CC (F.obj (op W)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map ιWU.op) : (a : CC (F.obj (op U))) → CC (F.obj (op W)))
            (surjInv ⋯ g) :
          CC (F.obj (op W)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map ιWV.op) : (a : CC (F.obj (op V))) → CC (F.obj (op W))) f :
          CC (F.obj (op W)))),
    ∀ᶠ (a : ↥U) in 𝓝 y,
      Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
        {g : F.EtaleSpace (CC := CC) (FC := FC) |
          ∃ (h : g.base (CC := CC) (FC := FC) ∈ V),
            g.germ (CC := CC) (FC := FC) =
              (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V (g.base (CC := CC) (FC := FC)) h) :
                  (a : CC (F.obj (op V))) → CC (F.stalk g.base))
                f}
        (mk (CC := CC) (FC := FC) (↑a : (↑X : Type v))
          ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (↑a : (↑X : Type v)) ⋯) :
              (a_1 : CC (F.obj (op U))) → CC (F.stalk (↑a : (↑X : Type v))))
            (surjInv ⋯ g)))
⊢ ∀ᶠ (a : ↥U) in 𝓝 y,
    Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
      {g : F.EtaleSpace (CC := CC) (FC := FC) |
        ∃ (h : g.base (CC := CC) (FC := FC) ∈ V),
          g.germ (CC := CC) (FC := FC) =
            (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V (g.base (CC := CC) (FC := FC)) h) :
                (a : CC (F.obj (op V))) → CC (F.stalk g.base))
              f}
      (mk (CC := CC) (FC := FC) (↑a : (↑X : Type v))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (↑a : (↑X : Type v)) ⋯) :
            (a_1 : CC (F.obj (op U))) → CC (F.stalk (↑a : (↑X : Type v))))
          (surjInv ⋯ g)))

Tactic:
rcases F.germ_eq _ _ _ _ _ hgf with ⟨W, hyW, ιWU, ιWV, hW⟩

Premises used: TopCat.Presheaf.germ_eq