-- Module: Mathlib.CategoryTheory.Galois.Topology
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_1 : GaloisCategory.{u₁, u₂} C
inst_2 : FiberFunctor F
S U : Set (Aut F)
hU : U ⊆ S
hUopen : IsOpen U
hUone : 1 ∈ U
I : Set C
w : Fintype (↑I : Type u₁)
hc : ∀ X ∈ I, IsConnected.{u₁, u₂} X
hmem : ∀ (σ : Aut F), (∀ (X : (↑I : Type u₁)), σ.hom.app (↑X : C) = 𝟙 (F.obj (↑X : C))) → σ ∈ U
P : C := piObj.{u₁, u₂, u₁} fun (X : (↑I : Type u₁)) ↦ (↑X : C)
A : C
a : (F.obj A).obj
hgal : IsGalois.{u₁, u₂} A
hbij :
  Function.Bijective.{u₂ + 1, w + 1} (β := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj P))
    fun (f : A ⟶ P) ↦
    (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
            (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
          _ _ _ (F.map f) :
        (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj P))
      a
t : Aut F
ht :
  (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
            (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
          _ _ _ (t.hom.app A) :
        (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A))
      a =
    a
this_1 : ∀ (X : (↑I : Type u₁)), IsConnected.{u₁, u₂} (↑X : C)
this : ∀ (X : (↑I : Type u₁)), Nonempty.{w + 1} (F.obj (↑X : C)).obj
X : (↑I : Type u₁)
x : (F.obj (↑X : C)).obj
refine_1 :
  ∀ (z : (F.obj (piObj.{u₁, u₂, u₁} fun (X : (↑I : Type u₁)) ↦ (↑X : C))).obj),
    Eq (α := (F.obj (↑X : C)).obj)
      ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
                (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
              _ _ _ (t.hom.app (↑X : C) : F.obj (↑X : C) ⟶ F.obj (↑X : C)) :
            (a : (F.obj (↑X : C)).obj) → (F.obj (↑X : C)).obj)
          ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
                  (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
                  (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
                  (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
                    (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
                  _ _ _
                  (F.map (Pi.π (fun (X : (↑I : Type u₁)) ↦ (↑X : C)) X) :
                    F.obj (piObj.{u₁, u₂, u₁} fun (X : (↑I : Type u₁)) ↦ (↑X : C)) ⟶ F.obj (↑X : C)) :
                (a :
                    (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
                      (F.obj (∏ᶜ fun (X : (↑I : Type u₁)) ↦ (↑X : C)))) →
                  (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (↑X : C)))
              z :
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (↑X : C))) :
        (F.obj (↑X : C)).obj)
      ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
                (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
              _ _ _
              (F.map (Pi.π (fun (X : (↑I : Type u₁)) ↦ (↑X : C)) X) :
                F.obj (piObj.{u₁, u₂, u₁} fun (X : (↑I : Type u₁)) ↦ (↑X : C)) ⟶ F.obj (↑X : C)) :
            (a :
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
                  (F.obj (∏ᶜ fun (X : (↑I : Type u₁)) ↦ (↑X : C)))) →
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (↑X : C)))
          z :
        (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (↑X : C)))
⊢ (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
            (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
          _ _ _ (t.hom.app (↑X : C) : F.obj (↑X : C) ⟶ F.obj (↑X : C)) :
        (a : (F.obj (↑X : C)).obj) → (F.obj (↑X : C)).obj)
      x =
    x

Tactic:
obtain ⟨z, rfl⟩ :=
        surjective_of_nonempty_fiber_of_isConnected F (Pi.π (fun X : I ↦ X.val) X) x

Premises used: CategoryTheory.Limits.Pi.π, CategoryTheory.PreGaloisCategory.surjective_of_nonempty_fiber_of_isConnected, Subtype.val, rfl