-- Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex
Goal:
n m : ℕ
f₁ : unop (op ⦋m⦌) ⟶ ⦋n⦌
h₁ :
  Membership.mem.{u, u} (α := Δ[n] _⦋m⦌) (γ := Set (Δ[n] _⦋m⦌)) (Δ[n].nonDegenerate m)
    ((Equiv.symm objEquiv : (a : unop (op ⦋m⦌) ⟶ ⦋n⦌) → Δ[n] _⦋m⦌) f₁ : Δ[n] _⦋m⦌)
f₂ : unop (op ⦋m⦌) ⟶ ⦋n⦌
h₂ :
  Membership.mem.{u, u} (α := Δ[n] _⦋m⦌) (γ := Set (Δ[n] _⦋m⦌)) (Δ[n].nonDegenerate m)
    ((Equiv.symm objEquiv : (a : unop (op ⦋m⦌) ⟶ ⦋n⦌) → Δ[n] _⦋m⦌) f₂ : Δ[n] _⦋m⦌)
h₃ :
  Eq (α :=
    { x : Finset (Fin (n + 1)) //
      Membership.mem (γ := Set (Finset (Fin (n + 1)))) {S : Finset (Fin (n + 1)) | #S = m + 1} x })
    ((fun (x : { x : Δ[n] _⦋m⦌ // Membership.mem (γ := Set (Δ[n] _⦋m⦌)) (Δ[n].nonDegenerate m) x }) ↦
        (match (motive :=
            { x : Δ[n] _⦋m⦌ // Membership.mem (γ := Set (Δ[n] _⦋m⦌)) (Δ[n].nonDegenerate m) x } →
              { x : Finset (Fin (n + 1)) //
                Membership.mem (γ := Set (Finset (Fin (n + 1)))) {S : Finset (Fin (n + 1)) | #S = m + 1} x })
            x with
          | ⟨x, hx⟩ =>
            Subtype.mk (p := fun (x : Finset (Fin (n + 1))) ↦
              Membership.mem (γ := Set (Finset (Fin (n + 1)))) {S : Finset (Fin (n + 1)) | #S = m + 1} x)
              (Finset.image (α := Fin ((unop (op ⦋m⦌)).len + 1))
                (⇑(Hom.toOrderHom
                      (((objEquiv : Δ[n] _⦋m⦌ ≃ (unop (op ⦋m⦌) ⟶ ⦋n⦌)) : (a : Δ[n] _⦋m⦌) → unop (op ⦋m⦌) ⟶ ⦋n⦌) x)) :
                  (a : Fin ((unop (op ⦋m⦌)).len + 1)) → Fin (⦋n⦌.len + 1))
                univ)
              ⋯ :
          { x : Finset (Fin (n + 1)) //
            Membership.mem (γ := Set (Finset (Fin (n + 1)))) {S : Finset (Fin (n + 1)) | #S = m + 1} x }))
      (Subtype.mk (p := fun (x : Δ[n] _⦋m⦌) ↦ Membership.mem (γ := Set (Δ[n] _⦋m⦌)) (Δ[n].nonDegenerate m) x)
        ((Equiv.symm objEquiv : (a : unop (op ⦋m⦌) ⟶ ⦋n⦌) → Δ[n] _⦋m⦌) f₁) h₁))
    ((fun (x : { x : Δ[n] _⦋m⦌ // Membership.mem (γ := Set (Δ[n] _⦋m⦌)) (Δ[n].nonDegenerate m) x }) ↦
        (match (motive :=
            { x : Δ[n] _⦋m⦌ // Membership.mem (γ := Set (Δ[n] _⦋m⦌)) (Δ[n].nonDegenerate m) x } →
              { x : Finset (Fin (n + 1)) //
                Membership.mem (γ := Set (Finset (Fin (n + 1)))) {S : Finset (Fin (n + 1)) | #S = m + 1} x })
            x with
          | ⟨x, hx⟩ =>
            Subtype.mk (p := fun (x : Finset (Fin (n + 1))) ↦
              Membership.mem (γ := Set (Finset (Fin (n + 1)))) {S : Finset (Fin (n + 1)) | #S = m + 1} x)
              (Finset.image (α := Fin ((unop (op ⦋m⦌)).len + 1))
                (⇑(Hom.toOrderHom
                      (((objEquiv : Δ[n] _⦋m⦌ ≃ (unop (op ⦋m⦌) ⟶ ⦋n⦌)) : (a : Δ[n] _⦋m⦌) → unop (op ⦋m⦌) ⟶ ⦋n⦌) x)) :
                  (a : Fin ((unop (op ⦋m⦌)).len + 1)) → Fin (⦋n⦌.len + 1))
                univ)
              ⋯ :
          { x : Finset (Fin (n + 1)) //
            Membership.mem (γ := Set (Finset (Fin (n + 1)))) {S : Finset (Fin (n + 1)) | #S = m + 1} x }))
      (Subtype.mk (p := fun (x : Δ[n] _⦋m⦌) ↦ Membership.mem (γ := Set (Δ[n] _⦋m⦌)) (Δ[n].nonDegenerate m) x)
        ((Equiv.symm objEquiv : (a : unop (op ⦋m⦌) ⟶ ⦋n⦌) → Δ[n] _⦋m⦌) f₂) h₂))
⊢ Function.Injective (α := Fin (m + 1)) (β := Fin (n + 1)) (⇑(Hom.toOrderHom f₁) : (a : Fin (m + 1)) → Fin (n + 1))

Tactic:
simp only [mem_nonDegenerate_iff_mono, Equiv.apply_symm_apply,
      SimplexCategory.mono_iff_injective, SimplexCategory.len_mk] at h₁ h₂

Premises used: SimplexCategory.len_mk, Equiv.apply_symm_apply, SimplexCategory.mono_iff_injective, SSet.stdSimplex.mem_nonDegenerate_iff_mono