Metamath Proof Explorer


Theorem cnptop2

Description: Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015)

Ref Expression
Assertion cnptop2 ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐾 ∈ Top )

Proof

Step Hyp Ref Expression
1 eqid 𝐽 = 𝐽
2 eqid 𝐾 = 𝐾
3 1 2 iscnp2 ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 𝐽 ) ∧ ( 𝐹 : 𝐽 𝐾 ∧ ∀ 𝑦𝐾 ( ( 𝐹𝑃 ) ∈ 𝑦 → ∃ 𝑥𝐽 ( 𝑃𝑥 ∧ ( 𝐹𝑥 ) ⊆ 𝑦 ) ) ) ) )
4 3 simplbi ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 𝐽 ) )
5 4 simp2d ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐾 ∈ Top )