-- Module: Mathlib.AlgebraicTopology.SimplicialSet.Skeleton
Goal:
X Y : SSet
i : X ⟶ Y
d : ℕ
x : SimplexCategoryᵒᵖ
n : ℕ
c₁ : Cell i d
f₁ : ⦋n⦌ ⟶ ⦋d⦌
w : Epi f₁
hx₁ :
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₁.ιSigmaStdSimplex.app (op ⦋n⦌)) :
          (a : (fun (X : Type u) ↦ X) (Δ[d] _⦋n⦌)) → (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌))
        ((Equiv.symm stdSimplex.objEquiv : (a : unop (op ⦋n⦌) ⟶ ⦋d⦌) → Δ[d] _⦋n⦌) f₁ : Δ[d] _⦋n⦌) :
      (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌)) ∉
    Set.range (ι := sigmaBoundary i d _⦋n⦌)
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) ((l i d).app (op ⦋n⦌))) :
        (a : sigmaBoundary i d _⦋n⦌) → sigmaStdSimplex i d _⦋n⦌)
f₂ : ⦋n⦌ ⟶ ⦋d⦌
w_1 : Epi f₂
hx₂ :
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₁.ιSigmaStdSimplex.app (op ⦋n⦌)) :
          (a : (fun (X : Type u) ↦ X) (Δ[d] _⦋n⦌)) → (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌))
        ((Equiv.symm stdSimplex.objEquiv : (a : unop (op ⦋n⦌) ⟶ ⦋d⦌) → Δ[d] _⦋n⦌) f₂ : Δ[d] _⦋n⦌) :
      (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌)) ∉
    Set.range (ι := sigmaBoundary i d _⦋n⦌)
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) ((l i d).app (op ⦋n⦌))) :
        (a : sigmaBoundary i d _⦋n⦌) → sigmaStdSimplex i d _⦋n⦌)
h :
  Eq (α := (fun (X : Type u) ↦ X) (Y _⦋n⦌))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (Y.map f₁.op) :
          (a : (fun (X : Type u) ↦ X) (Y _⦋d⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n⦌))
        c₁.simplex :
      (fun (X : Type u) ↦ X) (Y _⦋n⦌))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (Y.map f₂.op) :
          (a : (fun (X : Type u) ↦ X) (Y _⦋d⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n⦌))
        c₁.simplex :
      (fun (X : Type u) ↦ X) (Y _⦋n⦌))
refine_3 :
  Epi f₁ →
    ∀
      (hx₂ :
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₁.ιSigmaStdSimplex.app (op ⦋n⦌)) :
                (a : (fun (X : Type u) ↦ X) (Δ[d] _⦋n⦌)) → (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌))
              ((Equiv.symm stdSimplex.objEquiv : (a : unop (op ⦋n⦌) ⟶ ⦋d⦌) → Δ[d] _⦋n⦌) f₁ : Δ[d] _⦋n⦌) :
            (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌)) ∉
          Set.range (ι := sigmaBoundary i d _⦋n⦌)
            (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) ((l i d).app (op ⦋n⦌))) :
              (a : sigmaBoundary i d _⦋n⦌) → sigmaStdSimplex i d _⦋n⦌))
      (h :
        Eq (α := (fun (X : Type u) ↦ X) (Y _⦋n⦌))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (Y.map f₁.op) :
                (a : (fun (X : Type u) ↦ X) (Y _⦋d⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n⦌))
              c₁.simplex :
            (fun (X : Type u) ↦ X) (Y _⦋n⦌))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (Y.map f₁.op) :
                (a : (fun (X : Type u) ↦ X) (Y _⦋d⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n⦌))
              c₁.simplex :
            (fun (X : Type u) ↦ X) (Y _⦋n⦌))),
      Eq (α := sigmaStdSimplex i d _⦋n⦌)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₁.ιSigmaStdSimplex.app (op ⦋n⦌)) :
              (a : (fun (X : Type u) ↦ X) (Δ[d] _⦋n⦌)) → (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌))
            ((Equiv.symm stdSimplex.objEquiv : (a : unop (op ⦋n⦌) ⟶ ⦋d⦌) → Δ[d] _⦋n⦌) f₁ : Δ[d] _⦋n⦌) :
          (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₁.ιSigmaStdSimplex.app (op ⦋n⦌)) :
              (a : (fun (X : Type u) ↦ X) (Δ[d] _⦋n⦌)) → (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌))
            ((Equiv.symm stdSimplex.objEquiv : (a : unop (op ⦋n⦌) ⟶ ⦋d⦌) → Δ[d] _⦋n⦌) f₁ : Δ[d] _⦋n⦌) :
          (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌))
⊢ Eq (α := sigmaStdSimplex i d _⦋n⦌)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₁.ιSigmaStdSimplex.app (op ⦋n⦌)) :
          (a : (fun (X : Type u) ↦ X) (Δ[d] _⦋n⦌)) → (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌))
        ((Equiv.symm stdSimplex.objEquiv : (a : unop (op ⦋n⦌) ⟶ ⦋d⦌) → Δ[d] _⦋n⦌) f₁ : Δ[d] _⦋n⦌) :
      (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₁.ιSigmaStdSimplex.app (op ⦋n⦌)) :
          (a : (fun (X : Type u) ↦ X) (Δ[d] _⦋n⦌)) → (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌))
        ((Equiv.symm stdSimplex.objEquiv : (a : unop (op ⦋n⦌) ⟶ ⦋d⦌) → Δ[d] _⦋n⦌) f₂ : Δ[d] _⦋n⦌) :
      (fun (X : Type u) ↦ X) (sigmaStdSimplex i d _⦋n⦌))

Tactic:
obtain rfl : f₁ = f₂ :=
        Y.unique_nonDegenerate_map _ f₁ ⟨_, c₁.nonDegenerate⟩ rfl
          f₂ ⟨_, c₁.nonDegenerate⟩ h

Premises used: Subtype.mk, SSet.relativeCellComplexOfMono.Cell.nonDegenerate, SSet.unique_nonDegenerate_map, rfl