Metamath Proof Explorer


Theorem cntop2

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

Ref Expression
Assertion cntop2 ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐾 ∈ Top )

Proof

Step Hyp Ref Expression
1 eqid 𝐽 = 𝐽
2 eqid 𝐾 = 𝐾
3 1 2 iscn2 ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ) ∧ ( 𝐹 : 𝐽 𝐾 ∧ ∀ 𝑥𝐾 ( 𝐹𝑥 ) ∈ 𝐽 ) ) )
4 3 simplbi ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ) )
5 4 simprd ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐾 ∈ Top )