Theorem cnpf

Description: A continuous function at point P is a mapping. (Contributed by FL, 17-Nov-2006) (Revised by Mario Carneiro, 21-Aug-2015)

	Ref	Expression
Hypotheses	iscnp2.1	⊢ 𝑋 = ∪ 𝐽
	iscnp2.2	⊢ 𝑌 = ∪ 𝐾
Assertion	cnpf	⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐹 : 𝑋 ⟶ 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		iscnp2.1	⊢ 𝑋 = ∪ 𝐽
2		iscnp2.2	⊢ 𝑌 = ∪ 𝐾
3	1 2	iscnp2	⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋 ) ∧ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐾 ( ( 𝐹 ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) ) ) ) )
4	3	simprbi	⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐾 ( ( 𝐹 ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) ) ) )
5	4	simpld	⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐹 : 𝑋 ⟶ 𝑌 )