/-- One can reformulate smoothness within a set at a point as continuity within this set at this
point, and smoothness in the corresponding extended chart in the target. -/
theorem contMDiffWithinAt_iff_target :
    ContMDiffWithinAt I I' n f s x ↔
      ContinuousWithinAt f s x ∧ ContMDiffWithinAt I 𝓘(𝕜, E') n (extChartAt I' (f x) ∘ f) s x := by
  simp_rw [ContMDiffWithinAt, LiftPropWithinAt, ← and_assoc]
  have cont :
    ContinuousWithinAt f s x ∧ ContinuousWithinAt (extChartAt I' (f x) ∘ f) s x ↔
        ContinuousWithinAt f s x :=
      and_iff_left_of_imp <| (continuousAt_extChartAt _ _).comp_continuousWithinAt
  

Proof state:
𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x : M
m n : ℕ∞
cont : ContinuousWithinAt f s x ∧ ContinuousWithinAt (↑(extChartAt I' (f x)) ∘ f) s x ↔ ContinuousWithinAt f s x
⊢ ContinuousWithinAt f s x ∧
      ContDiffWithinAtProp I I' n (↑(chartAt H' (f x)) ∘ f ∘ ↑(PartialHomeomorph.symm (chartAt H x)))
        (↑(PartialHomeomorph.symm (chartAt H x)) ⁻¹' s) (↑(chartAt H x) x) ↔
    (ContinuousWithinAt f s x ∧ ContinuousWithinAt (↑(extChartAt I' (f x)) ∘ f) s x) ∧
      ContDiffWithinAtProp I 𝓘(𝕜, E') n
        (↑(chartAt E' ((↑(extChartAt I' (f x)) ∘ f) x)) ∘
          (↑(extChartAt I' (f x)) ∘ f) ∘ ↑(PartialHomeomorph.symm (chartAt H x)))
        (↑(PartialHomeomorph.symm (chartAt H x)) ⁻¹' s) (↑(chartAt H x) x)

Next tactic:
simp_rw [cont, ContDiffWithinAtProp, extChartAt, PartialHomeomorph.extend, PartialEquiv.coe_trans,
    ModelWithCorners.toPartialEquiv_coe, PartialHomeomorph.coe_coe, modelWithCornersSelf_coe,
    chartAt_self_eq, PartialHomeomorph.refl_apply, comp.left_id]