Theorem: PartialHomeomorph.Set.EqOn.restr_eqOn_source
File: Mathlib/Topology/PartialHomeomorph.lean

tactic: constructor
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✝ e e' : PartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ e.restr e'.source ≈ e'.restr e.source
state_after: case left
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✝ e e' : PartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ (e.restr e'.source).source = (e'.restr e.source).source

case right
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✝ e e' : PartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ EqOn (↑(e.restr e'.source)) (↑(e'.restr e.source)) (e.restr e'.source).source

tactic: rw [e'.restr_source' _ e.open_source]
state_before: case left
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✝ e e' : PartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ (e.restr e'.source).source = (e'.restr e.source).source
state_after: case left
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✝ e e' : PartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ (e.restr e'.source).source = e'.source ∩ e.source

tactic: rw [e.restr_source' _ e'.open_source]
state_before: case left
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✝ e e' : PartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ (e.restr e'.source).source = e'.source ∩ e.source
state_after: case left
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✝ e e' : PartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ e.source ∩ e'.source = e'.source ∩ e.source

tactic: exact Set.inter_comm _ _
state_before: case left
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✝ e e' : PartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ e.source ∩ e'.source = e'.source ∩ e.source
state_after: no goals

tactic: rw [e.restr_source' _ e'.open_source]
state_before: case right
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✝ e e' : PartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ EqOn (↑(e.restr e'.source)) (↑(e'.restr e.source)) (e.restr e'.source).source
state_after: case right
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✝ e e' : PartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ EqOn (↑(e.restr e'.source)) (↑(e'.restr e.source)) (e.source ∩ e'.source)

tactic: refine (EqOn.trans ?_ h).trans ?_ <;> simp only [mfld_simps, eqOn_refl]
state_before: case right
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✝ e e' : PartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ EqOn (↑(e.restr e'.source)) (↑(e'.restr e.source)) (e.source ∩ e'.source)
state_after: no goals