-- Module: Mathlib.Topology.Sheaves.LocalPredicate
Goal:
X : TopCat
T : (↑X : Type u_1) → Type u_1
P : LocalPredicate T
x : (↑X : Type u_1)
w :
  ∀ (U V : OpenNhds x) (fU : (y : ↥(↑U : Opens (↑X : Type u_1))) → T (↑y : (↑X : Type u_1))),
    P.toPrelocalPredicate (T := T).pred (T := T) fU →
      ∀ (fV : (y : ↥(↑V : Opens (↑X : Type u_1))) → T (↑y : (↑X : Type u_1))),
        P.toPrelocalPredicate (T := T).pred (T := T) fV →
          Eq (α :=
              T
                (↑(Subtype.mk (p := fun (x_3 : (↑X : Type u_1)) ↦ x_3 ∈ (↑U : Opens (↑X : Type u_1))) x ⋯) :
                  (↑X : Type u_1)))
              (fU (Subtype.mk (p := fun (x_3 : (↑X : Type u_1)) ↦ x_3 ∈ (↑U : Opens (↑X : Type u_1))) x ⋯))
              (fV (Subtype.mk (p := fun (x_3 : (↑X : Type u_1)) ↦ x_3 ∈ (↑V : Opens (↑X : Type u_1))) x ⋯)) →
            ∃ (W : OpenNhds x) (iU : W ⟶ U) (iV : W ⟶ V),
              ∀ (w : ↥(↑W : Opens (↑X : Type u_1))),
                Eq (α :=
                  T (↑((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑U : Opens (↑X : Type u_1))) w) : (↑X : Type u_1)))
                  (fU ((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑U : Opens (↑X : Type u_1))) w))
                  (fV ((iV : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑V : Opens (↑X : Type u_1))) w))
tU tV : (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))).stalk x
h :
  Eq (α := T x)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (stalkToFiber (T := T) P x) :
          (a : (subpresheafToTypes P.toPrelocalPredicate).stalk x) → T x)
        tU :
      T x)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (stalkToFiber (T := T) P x) :
          (a : (subpresheafToTypes P.toPrelocalPredicate).stalk x) → T x)
        tV :
      T x)
refine_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)
    (h :
      Eq (α := T x)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (stalkToFiber (T := T) P x) :
              (a : (subpresheafToTypes P.toPrelocalPredicate).stalk x) → T 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)) :
          T x)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (stalkToFiber (T := T) P x) :
              (a : (subpresheafToTypes P.toPrelocalPredicate).stalk x) → T x)
            tV :
          T x)),
    ∃ (W : (OpenNhds x)ᵒᵖ) (s : (w : ↥(↑(unop W) : Opens (↑X : Type u_1))) → T (↑w : (↑X : Type u_1))) (hW :
      P.toPrelocalPredicate (T := T).pred (T := T) s),
      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 W) : Opens (↑X : Type u_1)) x ⋯) :
                (a :
                    (fun (X : Type u_1) ↦ X)
                      ((subsheafToTypes P).presheaf.obj (op (↑(unop 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 W) : Opens (↑X : Type u_1))))) → T (↑x_1 : (↑X : Type u_1))) ↦
                P.toPrelocalPredicate (T := T).pred (T := T) f) s hW) :
            (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x)) ∧
        tV =
          (ConcreteCategory.hom (FC := TypeCat.Fun)
                ((subsheafToTypes (T := T) P).presheaf.germ (↑(unop W) : Opens (↑X : Type u_1)) x ⋯) :
              (a :
                  (fun (X : Type u_1) ↦ X)
                    ((subsheafToTypes P).presheaf.obj (op (↑(unop 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 W) : Opens (↑X : Type u_1))))) → T (↑x_1 : (↑X : Type u_1))) ↦
              P.toPrelocalPredicate (T := T).pred (T := T) f) s hW)
⊢ ∃ (W : (OpenNhds x)ᵒᵖ) (s : (w : ↥(↑(unop W) : Opens (↑X : Type u_1))) → T (↑w : (↑X : Type u_1))) (hW :
    P.toPrelocalPredicate (T := T).pred (T := T) s),
    tU =
        (ConcreteCategory.hom (FC := TypeCat.Fun)
              ((subsheafToTypes (T := T) P).presheaf.germ (↑(unop W) : Opens (↑X : Type u_1)) x ⋯) :
            (a : (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.obj (op (↑(unop 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 W) : Opens (↑X : Type u_1))))) → T (↑x_1 : (↑X : Type u_1))) ↦
            P.toPrelocalPredicate (T := T).pred (T := T) f) s hW) ∧
      tV =
        (ConcreteCategory.hom (FC := TypeCat.Fun)
              ((subsheafToTypes (T := T) P).presheaf.germ (↑(unop W) : Opens (↑X : Type u_1)) x ⋯) :
            (a : (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.obj (op (↑(unop 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 W) : Opens (↑X : Type u_1))))) → T (↑x_1 : (↑X : Type u_1))) ↦
            P.toPrelocalPredicate (T := T).pred (T := T) f) s hW)

Tactic:
obtain ⟨U, ⟨fU, hU⟩, rfl⟩ := jointly_surjective' tU

Premises used: CategoryTheory.Limits.Types.jointly_surjective', rfl