-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
inst_4 : HasLimits.{v, u} C
inst_5 : PreservesLimits (forget C (FC := FC) (CC := CC))
inst_6 : ReflectsIsomorphisms (forget C (FC := FC) (CC := CC))
F G : Sheaf C X
f : F ⟶ G
U : Opens (↑X : Type v)
hinj :
  ∀ x ∈ U,
    Function.Injective.{v + 1, v + 1} (α :=
      CC
        (obj (stalkFunctor C x)
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))))))
      (β :=
      CC
        (obj (stalkFunctor C x)
          (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))))))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f.hom)) :
        (a : CC ((stalkFunctor C x).obj F.obj)) → CC ((stalkFunctor C x).obj G.obj))
hsurj :
  ∀ (t : CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op U))),
    ∀ x ∈ U,
      ∃ (V : Opens (↑X : Type v)) (_ : x ∈ V) (iVU : V ⟶ U) (s :
        ToType (FC := FC) (CC := CC)
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V))),
        Eq (α :=
          CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
          ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.hom.app (op V)) :
                (a : CC (F.obj.obj (op V))) → CC (G.obj.obj (op V)))
              s :
            CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
          ((ConcreteCategory.hom (FC := FC) (CC := CC)
                  (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iVU.op) :
                (a : CC (G.obj.obj (op U))) → CC (G.obj.obj (op V)))
              t :
            CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
t : CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op U))
rw :
  ∀ (x : ↥U),
    ∃ (V : Opens (↑X : Type v)) (_ : (↑x : (↑X : Type v)) ∈ V) (iVU : V ⟶ U) (s :
      ToType (FC := FC) (CC := CC)
        (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V))),
      Eq (α := CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.hom.app (op V)) :
              (a : CC (F.obj.obj (op V))) → CC (G.obj.obj (op V)))
            s :
          CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC)
                (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iVU.op) :
              (a : CC (G.obj.obj (op U))) → CC (G.obj.obj (op V)))
            t :
          CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
| ∀ x ∈ U,
    ∃ (V : Opens (↑X : Type v)) (_ : x ∈ V) (iVU : V ⟶ U) (s :
      ToType (FC := FC) (CC := CC)
        (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V))),
      Eq (α := CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.hom.app (op V)) :
              (a : CC (F.obj.obj (op V))) → CC (G.obj.obj (op V)))
            s :
          CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC)
                (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iVU.op) :
              (a : CC (G.obj.obj (op U))) → CC (G.obj.obj (op V)))
            t :
          CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))

Tactic:
rw [Subtype.forall' (p := (· ∈ U))]

Premises used: Subtype.forall'