Theorem cnf

Description: A continuous function is a mapping. (Contributed by FL, 8-Dec-2006) (Revised by Mario Carneiro, 21-Aug-2015)

	Ref	Expression
Hypotheses	iscnp2.1	⊢ 𝑋 = ∪ 𝐽
	iscnp2.2	⊢ 𝑌 = ∪ 𝐾
Assertion	cnf	⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐹 : 𝑋 ⟶ 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		iscnp2.1	⊢ 𝑋 = ∪ 𝐽
2		iscnp2.2	⊢ 𝑌 = ∪ 𝐾
3	1 2	iscn2	⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ) ∧ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) )
4	3	simprbi	⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) )
5	4	simpld	⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐹 : 𝑋 ⟶ 𝑌 )