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

tactic: simp_rw [embedding_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
⊢ Embedding f ↔ ∀ (i : ι), Embedding ((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
⊢ Inducing f ∧ Function.Injective f ↔
    ∀ (i : ι), Inducing ((U i).carrier.restrictPreimage f) ∧ Function.Injective ((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
⊢ Inducing f ∧ Function.Injective f ↔
    ∀ (i : ι), Inducing ((U i).carrier.restrictPreimage f) ∧ Function.Injective ((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
⊢ Inducing f ∧ Function.Injective f ↔
    (∀ (x : ι), Inducing ((U x).carrier.restrictPreimage f)) ∧
      ∀ (x : ι), Function.Injective ((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
⊢ Inducing f ∧ Function.Injective f ↔
    (∀ (x : ι), Inducing ((U x).carrier.restrictPreimage f)) ∧
      ∀ (x : ι), Function.Injective ((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
⊢ Inducing f ↔ ∀ (x : ι), Inducing ((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
⊢ Function.Injective f ↔ ∀ (x : ι), Function.Injective ((U x).carrier.restrictPreimage f)

tactic: apply inducing_iff_inducing_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
⊢ Inducing f ↔ ∀ (x : ι), Inducing ((U x).carrier.restrictPreimage f)
state_after: no goals

tactic: apply Set.injective_iff_injective_of_iUnion_eq_univ
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
⊢ Function.Injective f ↔ ∀ (x : ι), Function.Injective ((U x).carrier.restrictPreimage f)
state_after: case h₂.hU
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
⊢ ⋃ i, (U i).carrier = univ

tactic: convert congr_arg SetLike.coe hU
state_before: case h₂.hU
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
⊢ ⋃ i, (U i).carrier = univ
state_after: case h.e'_2
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
⊢ ⋃ i, (U i).carrier = ↑(iSup U)

tactic: simp
state_before: case h.e'_2
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
⊢ ⋃ i, (U i).carrier = ↑(iSup U)
state_after: no goals