Theorem cntop1

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

	Ref	Expression
Assertion	cntop1	⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐽 ∈ 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 ∧ ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) )
4	3	simplbi	⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ) )
5	4	simpld	⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐽 ∈ Top )