-- Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex
Goal:
X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst : LinearOrder ι
f : P.RankFunction ι
inst_1 : P.IsProper
inst_2 : SuccOrder ι
inst_3 : NoMaxOrder ι
j : ι
x : SimplexCategoryᵒᵖ
d : ℕ
x₁ x₂ : Functor.obj (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌) (f.sigmaStdSimplex j)
hx₁ :
  x₁ ∉
    Set.range (ι :=
      (fun (X : Type u) ↦ X) (Functor.obj (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌) (f.sigmaHorn j)))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌) (f.m j))) :
        (a : (fun (X : Type u) ↦ X) (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌.obj (f.sigmaHorn j))) →
          (fun (X : Type u) ↦ X) (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌.obj (f.sigmaStdSimplex j)))
hx₂ :
  x₂ ∉
    Set.range (ι :=
      (fun (X : Type u) ↦ X) (Functor.obj (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌) (f.sigmaHorn j)))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌) (f.m j))) :
        (a : (fun (X : Type u) ↦ X) (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌.obj (f.sigmaHorn j))) →
          (fun (X : Type u) ↦ X) (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌.obj (f.sigmaStdSimplex j)))
h :
  Eq (α :=
    (fun (X : Type u) ↦ X)
      (Functor.obj (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌) (f.filtration (Order.succ j)).toSSet))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌) (f.b j)) :
          (a :
              (fun (X : Type u) ↦ X)
                (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌.obj (f.sigmaStdSimplex j))) →
            (fun (X : Type u) ↦ X)
              (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌.obj (f.filtration (Order.succ j)).toSSet))
        x₁ :
      (fun (X : Type u) ↦ X)
        (Functor.obj (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌) (f.filtration (Order.succ j)).toSSet))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌) (f.b j)) :
          (a :
              (fun (X : Type u) ↦ X)
                (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌.obj (f.sigmaStdSimplex j))) →
            (fun (X : Type u) ↦ X)
              (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌.obj (f.filtration (Order.succ j)).toSSet))
        x₂ :
      (fun (X : Type u) ↦ X)
        (Functor.obj (CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌) (f.filtration (Order.succ j)).toSSet))
s₁ : (range (X := f.sigmaHorn j) (f.m j)).N
g₁ : ⦋d⦌ ⟶ ⦋s₁.dim⦌
left : Epi g₁
hg₁ :
  (ConcreteCategory.hom (FC := TypeCat.Fun) ((f.sigmaStdSimplex j).map g₁.op) :
        (a : (fun (X : Type u) ↦ X) (f.sigmaStdSimplex j _⦋s₁.dim⦌)) →
          (fun (X : Type u) ↦ X) (f.sigmaStdSimplex j _⦋d⦌))
      s₁.simplex =
    x₁
g₂ : ⦋d⦌ ⟶ ⦋s₁.dim⦌
left_1 : Epi g₂
hg₂ :
  (ConcreteCategory.hom (FC := TypeCat.Fun) ((f.sigmaStdSimplex j).map g₂.op) :
        (a : (fun (X : Type u) ↦ X) (f.sigmaStdSimplex j _⦋s₁.dim⦌)) →
          (fun (X : Type u) ↦ X) (f.sigmaStdSimplex j _⦋d⦌))
      s₁.simplex =
    x₂
⊢ Eq (α := X _⦋d⦌)
    ((↑((ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app (f.b j) (op ⦋d⦌)) :
              (a : (fun (X : Type u) ↦ X) (f.sigmaStdSimplex j _⦋d⦌)) →
                (fun (X : Type u) ↦ X) ((f.filtration (Order.succ j)).toSSet _⦋d⦌))
            x₁) :
        X _⦋d⦌) :
      X _⦋d⦌)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map g₂.op) :
          (a : (fun (X : Type u) ↦ X) (X _⦋s₁.dim⦌)) → (fun (X : Type u) ↦ X) (X _⦋d⦌))
        (↑(Subtype.mk (p := fun (x : X _⦋s₁.dim⦌) ↦ Membership.mem (γ := Set (X _⦋s₁.dim⦌)) (X.nonDegenerate s₁.dim) x)
              (f.mapN s₁).simplex ⋯) :
          X _⦋s₁.dim⦌) :
      (fun (X : Type u) ↦ X) (X _⦋d⦌))

Tactic:
simp [mapN, dsimp% h, ← hg₂, dsimp% NatTrans.naturality_apply (f.b j)]

Premises used: CategoryTheory.NatTrans.naturality_apply, SSet.Subcomplex.Pairing.RankFunction.b, SSet.Subcomplex.Pairing.RankFunction.mapN