Theorem: openEmbedding_iff_openEmbedding_of_iSup_eq_top
File: Mathlib/Topology/LocalAtTarget.lean

tactic: simp_rw [openEmbedding_iff]
state_before: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ OpenEmbedding f ↔ ∀ (i : ι), OpenEmbedding ((U i).carrier.restrictPreimage f)
state_after: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Embedding f ∧ IsOpen (range f) ↔
    ∀ (i : ι), Embedding ((U i).carrier.restrictPreimage f) ∧ IsOpen (range ((U i).carrier.restrictPreimage f))

tactic: rw [forall_and]
state_before: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Embedding f ∧ IsOpen (range f) ↔
    ∀ (i : ι), Embedding ((U i).carrier.restrictPreimage f) ∧ IsOpen (range ((U i).carrier.restrictPreimage f))
state_after: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Embedding f ∧ IsOpen (range f) ↔
    (∀ (x : ι), Embedding ((U x).carrier.restrictPreimage f)) ∧
      ∀ (x : ι), IsOpen (range ((U x).carrier.restrictPreimage f))

tactic: apply and_congr
state_before: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Embedding f ∧ IsOpen (range f) ↔
    (∀ (x : ι), Embedding ((U x).carrier.restrictPreimage f)) ∧
      ∀ (x : ι), IsOpen (range ((U x).carrier.restrictPreimage f))
state_after: case h₁
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Embedding f ↔ ∀ (x : ι), Embedding ((U x).carrier.restrictPreimage f)

case h₂
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ IsOpen (range f) ↔ ∀ (x : ι), IsOpen (range ((U x).carrier.restrictPreimage f))

tactic: apply embedding_iff_embedding_of_iSup_eq_top <;> assumption
state_before: case h₁
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Embedding f ↔ ∀ (x : ι), Embedding ((U x).carrier.restrictPreimage f)
state_after: no goals

tactic: simp_rw [Set.range_restrictPreimage]
state_before: case h₂
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ IsOpen (range f) ↔ ∀ (x : ι), IsOpen (range ((U x).carrier.restrictPreimage f))
state_after: case h₂
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ IsOpen (range f) ↔ ∀ (x : ι), IsOpen (Subtype.val ⁻¹' range f)

tactic: apply isOpen_iff_coe_preimage_of_iSup_eq_top hU
state_before: case h₂
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ IsOpen (range f) ↔ ∀ (x : ι), IsOpen (Subtype.val ⁻¹' range f)
state_after: no goals