case h
α : Type u_1
β : Type u_2
γ : Type ?u.5388
δ : Type ?u.5391
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
hf : Inducing f
s : Set α
x : α
⊢ x ∈ closure s ↔ x ∈ f ⁻¹' closure (f '' s)
α : Type u_1
β : Type u_2
γ : Type ?u.5388
δ : Type ?u.5391
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
hf : Inducing f
s : Set α
⊢ closure s = f ⁻¹' closure (f '' s)
ext x
no goals
case h
α : Type u_1
β : Type u_2
γ : Type ?u.5388
δ : Type ?u.5391
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
hf : Inducing f
s : Set α
x : α
⊢ x ∈ closure s ↔ x ∈ f ⁻¹' closure (f '' s)
rw [Set.mem_preimage, ← closure_induced, hf.induced]