/-- Being differentiable in the model space is a local property, invariant under smooth maps.
Therefore, it will lift nicely to manifolds. -/
theorem differentiable_within_at_localInvariantProp :
    (contDiffGroupoid ⊤ I).LocalInvariantProp (contDiffGroupoid ⊤ I')
      (DifferentiableWithinAtProp I I') :=
  { is_local := by
      intro s x u f u_open xu
      have : I.symm ⁻¹' (s ∩ u) ∩ Set.range I = I.symm ⁻¹' s ∩ Set.range I ∩ I.symm ⁻¹' u := by
        simp only [Set.inter_right_comm, Set.preimage_inter]
      rw [DifferentiableWithinAtProp, DifferentiableWithinAtProp, this]
      symm
      apply differentiableWithinAt_inter
      have : u ∈ 𝓝 (I.symm (I x)) := by
        rw [ModelWithCorners.left_inv]
        exact IsOpen.mem_nhds u_open xu
      apply I.continuous_symm.continuousAt this
    right_invariance' := by
      intro s x f e he hx h
      rw [DifferentiableWithinAtProp] at h ⊢
      have : I x = (I ∘ e.symm ∘ I.symm) (I (e x)) := by simp only [hx, mfld_simps]
      rw [this] at h
      have : I (e x) ∈ I.symm ⁻¹' e.target ∩ Set.range I := by simp only [hx, mfld_simps]
      have := (mem_groupoid_of_pregroupoid.2 he).2.contDiffWithinAt this
      convert (h.comp' _ (this.differentiableWithinAt le_top)).mono_of_mem _ using 1
      · ext y; simp only [mfld_simps]
      refine'
        mem_nhdsWithin.mpr
          ⟨I.symm ⁻¹' e.target, e.open_target.preimage I.continuous_symm, by
            simp_rw [Set.mem_preimage, I.left_inv, e.mapsTo hx], _⟩
      mfld_set_tac
    congr_of_forall := by
      intro s x f g h hx hf
      apply hf.congr
      · intro y hy
        simp only [mfld_simps] at hy
        simp only [h, hy, mfld_simps]
      · simp only [hx, mfld_simps]
    left_invariance' := by
      intro s x f e' he' hs hx h
      rw [DifferentiableWithinAtProp] at h ⊢
      have A : (I' ∘ f ∘ I.symm) (I x) ∈ I'.symm ⁻¹' e'.source ∩ Set.range I' := by
        simp only [hx, mfld_simps]
      have := (mem_groupoid_of_pregroupoid.2 he').1.contDiffWithinAt A
      convert (this.differentiableWithinAt le_top).comp _ h _
      · ext y; simp only [mfld_simps]
      · intro y hy; simp only [mfld_simps] at hy; 

Proof state:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
s : Set H
x : H
f : H → H'
e' : PartialHomeomorph H' H'
he' : e' ∈ contDiffGroupoid ⊤ I'
hs : s ⊆ f ⁻¹' e'.source
hx : f x ∈ e'.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I) (↑I x)
A : (↑I' ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑I x) ∈ ↑(ModelWithCorners.symm I') ⁻¹' e'.source ∩ range ↑I'
this :
  ContDiffWithinAt 𝕜 ⊤ (↑I' ∘ ↑e' ∘ ↑(ModelWithCorners.symm I')) (↑(ModelWithCorners.symm I') ⁻¹' e'.source ∩ range ↑I')
    ((↑I' ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑I x))
y : E
hy : ↑(ModelWithCorners.symm I) y ∈ s ∧ y ∈ range ↑I
⊢ (↑I' ∘ f ∘ ↑(ModelWithCorners.symm I)) y ∈ ↑(ModelWithCorners.symm I') ⁻¹' e'.source ∩ range ↑I'

Next tactic:
simpa only [hy, mfld_simps] using hs hy.1