-- 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⦌)
c₂ : Cell i d
f₂ : ⦋n⦌ ⟶ ⦋d⦌
w_1 : Epi f₂
hx₂ :
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₂.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.{u + 1} (α := Y _⦋n⦌)
    (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((b i d).app (op ⦋n⦌)) :
            (a : sigmaStdSimplex i d _⦋n⦌) →
              (↑(((skeletonOfMono i : (a : ℕ) → Y.Subcomplex) (d + 1)).obj (op ⦋n⦌)) : Type u))
          ((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⦌))) :
      Y _⦋n⦌)
    (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((b i d).app (op ⦋n⦌)) :
            (a : sigmaStdSimplex i d _⦋n⦌) →
              (↑(((skeletonOfMono i : (a : ℕ) → Y.Subcomplex) (d + 1)).obj (op ⦋n⦌)) : Type u))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₂.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⦌))) :
      Y _⦋n⦌)
rwa :
  Eq (α := (fun (X : Type u) ↦ X) (Y _⦋n⦌))
    ((↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((b i d).app (op ⦋n⦌)) :
              (a : sigmaStdSimplex i d _⦋n⦌) →
                (↑(((skeletonOfMono i : (a : ℕ) → Y.Subcomplex) (d + 1)).obj (op ⦋n⦌)) : Type u))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₁.ιSigmaStdSimplex.app (op ⦋n⦌)) :
                (a : Δ[d] _⦋n⦌) → sigmaStdSimplex i d _⦋n⦌)
              ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n⦌ ⟶ ⦋d⦌) → Δ[d] _⦋n⦌) f₁ : Δ[d] _⦋n⦌))) :
        Y _⦋n⦌) :
      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⦌))
        c₂.simplex :
      (fun (X : Type u) ↦ X) (Y _⦋n⦌))
rwa₁ rwa₂ rwa₃ rwa₄ :
  Eq.{u + 1} (α := (fun (X : Type u) ↦ X) (Y _⦋n⦌))
    (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((b i d).app (op ⦋n⦌)) :
            (a : sigmaStdSimplex i d _⦋n⦌) →
              (↑(((skeletonOfMono i : (a : ℕ) → Y.Subcomplex) (d + 1)).obj (op ⦋n⦌)) : Type u))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₁.ιSigmaStdSimplex.app (op ⦋n⦌)) :
              (a : Δ[d] _⦋n⦌) → sigmaStdSimplex i d _⦋n⦌)
            ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n⦌ ⟶ ⦋d⦌) → Δ[d] _⦋n⦌) f₁ : Δ[d] _⦋n⦌))) :
      Y _⦋n⦌)
    (↑((ConcreteCategory.hom (FC := TypeCat.Fun) ((b i d).app (op ⦋n⦌)) :
            (a : sigmaStdSimplex i d _⦋n⦌) →
              (↑(((skeletonOfMono i : (a : ℕ) → Y.Subcomplex) (d + 1)).obj (op ⦋n⦌)) : Type u))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (c₂.ιSigmaStdSimplex.app (op ⦋n⦌)) :
              (a : Δ[d] _⦋n⦌) → sigmaStdSimplex i d _⦋n⦌)
            ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n⦌ ⟶ ⦋d⦌) → Δ[d] _⦋n⦌) f₂ : Δ[d] _⦋n⦌))) :
      Y _⦋n⦌)
⊢ 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⦌))
        c₂.simplex :
      (fun (X : Type u) ↦ X) (Y _⦋n⦌))

Tactic:
rwa [← c₁.b_app_ι_app_objEquiv_symm_val f₁, ← c₂.b_app_ι_app_objEquiv_symm_val f₂]

Premises used: SSet.relativeCellComplexOfMono.Cell.b_app_ι_app_objEquiv_symm_val