Theorem: PartialHomeomorph.continuousOn_iff_continuousOn_comp_right
File: Mathlib/Topology/PartialHomeomorph.lean

Proof steps:
state_before: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : Y → Z
s : Set Y
h : s ⊆ e.target
⊢ ContinuousOn f s ↔ ContinuousOn (f ∘ ↑e) (e.source ∩ ↑e ⁻¹' s)
state_after: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : Y → Z
s : Set Y
h : s ⊆ e.target
⊢ (∀ x ∈ s, ContinuousWithinAt f s x) ↔ ∀ ⦃x : Y⦄, x ∈ s → ContinuousWithinAt (f ∘ ↑e) (↑e.symm '' s) (↑e.symm x)
tactic: simp only [← e.symm_image_eq_source_inter_preimage h, ContinuousOn, forall_mem_image]
state_before: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : Y → Z
s : Set Y
h : s ⊆ e.target
⊢ (∀ x ∈ s, ContinuousWithinAt f s x) ↔ ∀ ⦃x : Y⦄, x ∈ s → ContinuousWithinAt (f ∘ ↑e) (↑e.symm '' s) (↑e.symm x)
state_after: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : Y → Z
s : Set Y
h : s ⊆ e.target
x : Y
hx : x ∈ s
⊢ ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ∘ ↑e) (↑e.symm '' s) (↑e.symm x)
tactic: refine forall₂_congr fun x hx => ?_
state_before: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : Y → Z
s : Set Y
h : s ⊆ e.target
x : Y
hx : x ∈ s
⊢ ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ∘ ↑e) (↑e.symm '' s) (↑e.symm x)
state_after: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : Y → Z
s : Set Y
h : s ⊆ e.target
x : Y
hx : x ∈ s
⊢ e.source ∈ 𝓝 (↑e.symm x)
tactic: rw [e.continuousWithinAt_iff_continuousWithinAt_comp_right (h hx),
    e.symm_image_eq_source_inter_preimage h, inter_comm, continuousWithinAt_inter]
state_before: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : Y → Z
s : Set Y
h : s ⊆ e.target
x : Y
hx : x ∈ s
⊢ e.source ∈ 𝓝 (↑e.symm x)
state_after: no goals
tactic: exact IsOpen.mem_nhds e.open_source (e.map_target (h hx))