/-- If a function is `C^n` with `1 ≤ n` on a domain with unique derivatives, then its bundled
derivative is continuous. In this auxiliary lemma, we prove this fact when the source and target
space are model spaces in models with corners. The general fact is proved in
`ContMDiffOn.continuousOn_tangentMapWithin`-/
theorem ContMDiffOn.continuousOn_tangentMapWithin_aux {f : H → H'} {s : Set H}
    (hf : ContMDiffOn I I' n f s) (hn : 1 ≤ n) (hs : UniqueMDiffOn I s) :
    ContinuousOn (tangentMapWithin I I' f s) (π E (TangentSpace I) ⁻¹' s) := by
  suffices h :
    ContinuousOn
      (fun p : H × E =>
        (f p.fst,
          (fderivWithin 𝕜 (writtenInExtChartAt I I' p.fst f) (I.symm ⁻¹' s ∩ range I)
                ((extChartAt I p.fst) p.fst) : E →L[𝕜] E') p.snd)) (Prod.fst ⁻¹' s)
  · have A := (tangentBundleModelSpaceHomeomorph H I).continuous
    rw [continuous_iff_continuousOn_univ] at A
    have B :=
      ((tangentBundleModelSpaceHomeomorph H' I').symm.continuous.comp_continuousOn h).comp' A
    have :
      univ ∩ tangentBundleModelSpaceHomeomorph H I ⁻¹' (Prod.fst ⁻¹' s) =
        π E (TangentSpace I) ⁻¹' s :=
      by ext ⟨x, v⟩; simp only [mfld_simps]
    rw [this] at B
    apply B.congr
    rintro ⟨x, v⟩ hx
    dsimp [tangentMapWithin]
    ext; · rfl
    simp only [mfld_simps]
    apply congr_fun
    apply congr_arg
    rw [MDifferentiableWithinAt.mfderivWithin (hf.mdifferentiableOn hn x hx)]
    rfl
  suffices h :
    ContinuousOn
      (fun p : H × E =>
        (fderivWithin 𝕜 (I' ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I p.fst) : E →L[𝕜] E') p.snd)
      (Prod.fst ⁻¹' s)
  · dsimp [writtenInExtChartAt, extChartAt]
    exact (ContinuousOn.comp hf.continuousOn continuous_fst.continuousOn Subset.rfl).prod h
  suffices h : ContinuousOn (fderivWithin 𝕜 (I' ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I)) (I '' s)
  · have C := ContinuousOn.comp h I.continuous_toFun.continuousOn Subset.rfl
    have A : Continuous fun q : (E →L[𝕜] E') × E => q.1 q.2 :=
      isBoundedBilinearMap_apply.continuous
    have B :
      ContinuousOn
        (fun p : H × E => (fderivWithin 𝕜 (I' ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I p.1), p.2))
        (Prod.fst ⁻¹' s) := by
      apply ContinuousOn.prod _ continuous_snd.continuousOn
      refine C.comp continuousOn_fst ?_
      

Proof state:
𝕜 : Type u_1
inst✝²⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace 𝕜 E
H : Type u_3
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
Is : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁸ : NormedAddCommGroup E'
inst✝¹⁷ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝¹⁶ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹⁵ : TopologicalSpace M'
inst✝¹⁴ : ChartedSpace H' M'
I's : SmoothManifoldWithCorners I' M'
F : Type u_8
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜 F
G : Type u_9
inst✝¹¹ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹⁰ : TopologicalSpace N
inst✝⁹ : ChartedSpace G N
Js : SmoothManifoldWithCorners J N
F' : Type u_11
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
G' : Type u_12
inst✝⁶ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_13
inst✝⁵ : TopologicalSpace N'
inst✝⁴ : ChartedSpace G' N'
J's : SmoothManifoldWithCorners J' N'
F₁ : Type u_14
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_15
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
f✝ f₁ : M → M'
s✝ s₁ t : Set M
x : M
m n : ℕ∞
f : H → H'
s : Set H
hf : ContMDiffOn I I' n f s
hn : 1 ≤ n
hs : UniqueMDiffOn I s
h :
  ContinuousOn (fderivWithin 𝕜 (↑I' ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))
    (↑I '' s)
C :
  ContinuousOn
    (fderivWithin 𝕜 (↑I' ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I) ∘
      ↑I.toPartialEquiv)
    fun x => (↑I '' s) (↑I.toPartialEquiv x)
A : Continuous fun q => q.1 q.2
⊢ MapsTo (fun x => x.1) (Prod.fst ⁻¹' s) fun x => (↑I '' s) (↑I.toPartialEquiv x)

Next tactic:
exact preimage_mono (subset_preimage_image _ _)