α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type ?u.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Inducing f ∧ Function.Injective f ↔
    ∀ (i : ι), Inducing (restrictPreimage (U i).carrier f) ∧ Function.Injective (restrictPreimage (U i).carrier f)
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type ?u.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Embedding f ↔ ∀ (i : ι), Embedding (restrictPreimage (U i).carrier f)
simp_rw [embedding_iff]
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type ?u.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Inducing f ∧ Function.Injective f ↔
    (∀ (x : ι), Inducing (restrictPreimage (U x).carrier f)) ∧
      ∀ (x : ι), Function.Injective (restrictPreimage (U x).carrier f)
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type ?u.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Inducing f ∧ Function.Injective f ↔
    ∀ (i : ι), Inducing (restrictPreimage (U i).carrier f) ∧ Function.Injective (restrictPreimage (U i).carrier f)
rw [forall_and]
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.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Inducing f ↔ ∀ (x : ι), Inducing (restrictPreimage (U x).carrier 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.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Function.Injective f ↔ ∀ (x : ι), Function.Injective (restrictPreimage (U x).carrier f)
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
ι : Type ?u.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Inducing f ∧ Function.Injective f ↔
    (∀ (x : ι), Inducing (restrictPreimage (U x).carrier f)) ∧
      ∀ (x : ι), Function.Injective (restrictPreimage (U x).carrier f)
apply and_congr
no goals
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.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Inducing f ↔ ∀ (x : ι), Inducing (restrictPreimage (U x).carrier f)
apply inducing_iff_inducing_of_iSup_eq_top <;> assumption
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.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ (⋃ (i : ι), (U i).carrier) = univ
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.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ Function.Injective f ↔ ∀ (x : ι), Function.Injective (restrictPreimage (U x).carrier f)
apply Set.injective_iff_injective_of_iUnion_eq_univ
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.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ (⋃ (i : ι), (U i).carrier) = ↑(iSup U)
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.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ (⋃ (i : ι), (U i).carrier) = univ
convert congr_arg SetLike.coe hU
no goals
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.40608
U : ι → Opens β
hU : iSup U = ⊤
h : Continuous f
⊢ (⋃ (i : ι), (U i).carrier) = ↑(iSup U)
simp