-- Module: Mathlib.Topology.Sheaves.LocalPredicate
Goal:
X : TopCat
T : (↑X : Type u_1) → Type u_1
P : LocalPredicate T
x : (↑X : Type u_1)
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T (↑x_1 : (↑X : Type u_1))
hU : P.toPrelocalPredicate (T := T).pred (T := T) fU
V : (OpenNhds x)ᵒᵖ
fV : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T (↑x_1 : (↑X : Type u_1))
hV : P.toPrelocalPredicate (T := T).pred (T := T) fV
h :
  Eq (α := T x)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((P.cocone (T := T) x).ι.app U) :
          (a : { f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop U))) → T (↑x_1 : (↑X : Type u_1)) // P.pred f }) →
            (P.cocone x).pt)
        (Subtype.mk (p := fun (f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop U))) → T (↑x_1 : (↑X : Type u_1))) ↦
          P.toPrelocalPredicate (T := T).pred (T := T) f) fU hU) :
      (P.cocone (T := T) x).pt)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((P.cocone (T := T) x).ι.app V) :
          (a : { f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop V))) → T (↑x_1 : (↑X : Type u_1)) // P.pred f }) →
            (P.cocone x).pt)
        (Subtype.mk (p := fun (f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop V))) → T (↑x_1 : (↑X : Type u_1))) ↦
          P.toPrelocalPredicate (T := T).pred (T := T) f) fV hV) :
      (P.cocone (T := T) x).pt)
W : OpenNhds x
iU : W ⟶ unop U
iV : W ⟶ unop V
w :
  ∀ (w : ↥(↑W : Opens (↑X : Type u_1))),
    Eq (α :=
      T (↑((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑(unop U) : Opens (↑X : Type u_1))) w) : (↑X : Type u_1)))
      (fU ((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑(unop U) : Opens (↑X : Type u_1))) w))
      (fV ((iV : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑(unop V) : Opens (↑X : Type u_1))) w))
⊢ Eq (α := (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (colimit.ι
                (Functor.obj
                  (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                    (OpenNhds.inclusion x).op)
                  (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))
                U) :
            (a :
                (fun (X : Type u_1) ↦ X)
                  ((((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                            (OpenNhds.inclusion x).op).obj
                        (subpresheafToTypes P.toPrelocalPredicate)).obj
                    U)) →
              (fun (X : Type u_1) ↦ X)
                (colimit
                  (((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                        (OpenNhds.inclusion x).op).obj
                    (subpresheafToTypes P.toPrelocalPredicate))))
          (Subtype.mk (p := fun (f : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T (↑x_1 : (↑X : Type u_1))) ↦
            P.toPrelocalPredicate (T := T).pred (T := T) f) fU hU) :
        (fun (X : Type u_1) ↦ X)
          (colimit
            (Functor.obj
              (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                (OpenNhds.inclusion x).op)
              (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              ((subsheafToTypes (T := T) P).presheaf.germ (↑(unop (op W)) : Opens (↑X : Type u_1)) x ⋯) :
            (a :
                (fun (X : Type u_1) ↦ X)
                  ((subsheafToTypes P).presheaf.obj (op (↑(unop (op W)) : Opens (↑X : Type u_1))))) →
              (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.stalk x))
          (Subtype.mk (p :=
            fun (f : (x_1 : ↥(unop (op (↑(unop (op W)) : Opens (↑X : Type u_1))))) → T (↑x_1 : (↑X : Type u_1))) ↦
            P.toPrelocalPredicate (T := T).pred (T := T) f)
            (fun (w : ↥(↑(unop (op W)) : Opens (↑X : Type u_1))) ↦
              fU ((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑(unop U) : Opens (↑X : Type u_1))) w))
            ⋯) :
        (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x)) ∧
    Eq (α := (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (colimit.ι
                (Functor.obj
                  (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                    (OpenNhds.inclusion x).op)
                  (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))
                V) :
            (a :
                (fun (X : Type u_1) ↦ X)
                  ((((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                            (OpenNhds.inclusion x).op).obj
                        (subpresheafToTypes P.toPrelocalPredicate)).obj
                    V)) →
              (fun (X : Type u_1) ↦ X)
                (colimit
                  (((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                        (OpenNhds.inclusion x).op).obj
                    (subpresheafToTypes P.toPrelocalPredicate))))
          (Subtype.mk (p := fun (f : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T (↑x_1 : (↑X : Type u_1))) ↦
            P.toPrelocalPredicate (T := T).pred (T := T) f) fV hV) :
        (fun (X : Type u_1) ↦ X)
          (colimit
            (Functor.obj
              (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                (OpenNhds.inclusion x).op)
              (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              ((subsheafToTypes (T := T) P).presheaf.germ (↑(unop (op W)) : Opens (↑X : Type u_1)) x ⋯) :
            (a :
                (fun (X : Type u_1) ↦ X)
                  ((subsheafToTypes P).presheaf.obj (op (↑(unop (op W)) : Opens (↑X : Type u_1))))) →
              (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.stalk x))
          (Subtype.mk (p :=
            fun (f : (x_1 : ↥(unop (op (↑(unop (op W)) : Opens (↑X : Type u_1))))) → T (↑x_1 : (↑X : Type u_1))) ↦
            P.toPrelocalPredicate (T := T).pred (T := T) f)
            (fun (w : ↥(↑(unop (op W)) : Opens (↑X : Type u_1))) ↦
              fU ((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑(unop U) : Opens (↑X : Type u_1))) w))
            ⋯) :
        (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x))

Tactic:
⟨colimit_sound iU.op (Subtype.ext rfl), colimit_sound iV.op (Subtype.ext (funext w).symm)⟩

Premises used: Quiver.Hom.op, CategoryTheory.Limits.Types.colimit_sound, funext, And.intro, Eq.symm, rfl, Subtype.ext