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

tactic: refine ⟨fun h ↦ ?_, fun h ↦ e.symm.isOpen_image_of_subset_source h hs⟩
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
t : Set Y
hs : t ⊆ e.target
⊢ IsOpen (↑e.symm '' t) ↔ IsOpen t
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
t : Set Y
hs : t ⊆ e.target
h : IsOpen (↑e.symm '' t)
⊢ IsOpen t

tactic: have hs' : e.symm '' t ⊆ e.source := by
  rw [e.symm_image_eq_source_inter_preimage hs]
  apply Set.inter_subset_left
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
t : Set Y
hs : t ⊆ e.target
h : IsOpen (↑e.symm '' t)
⊢ IsOpen t
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
t : Set Y
hs : t ⊆ e.target
h : IsOpen (↑e.symm '' t)
hs' : ↑e.symm '' t ⊆ e.source
⊢ IsOpen t

tactic: rw [← e.image_symm_image_of_subset_target hs]
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
t : Set Y
hs : t ⊆ e.target
h : IsOpen (↑e.symm '' t)
hs' : ↑e.symm '' t ⊆ e.source
⊢ IsOpen t
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
t : Set Y
hs : t ⊆ e.target
h : IsOpen (↑e.symm '' t)
hs' : ↑e.symm '' t ⊆ e.source
⊢ IsOpen (↑e.toPartialEquiv '' (↑e.symm '' t))

tactic: exact e.isOpen_image_of_subset_source h hs'
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
t : Set Y
hs : t ⊆ e.target
h : IsOpen (↑e.symm '' t)
hs' : ↑e.symm '' t ⊆ e.source
⊢ IsOpen (↑e.toPartialEquiv '' (↑e.symm '' t))
state_after: no goals

tactic: rw [e.symm_image_eq_source_inter_preimage hs]
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
t : Set Y
hs : t ⊆ e.target
h : IsOpen (↑e.symm '' t)
⊢ ↑e.symm '' t ⊆ e.source
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
t : Set Y
hs : t ⊆ e.target
h : IsOpen (↑e.symm '' t)
⊢ e.source ∩ ↑e ⁻¹' t ⊆ e.source

tactic: apply Set.inter_subset_left
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
t : Set Y
hs : t ⊆ e.target
h : IsOpen (↑e.symm '' t)
⊢ e.source ∩ ↑e ⁻¹' t ⊆ e.source
state_after: no goals