Metamath Proof Explorer


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 : 𝑋𝑌 )

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 ∧ 𝑃𝑋 ) ∧ ( 𝐹 : 𝑋𝑌 ∧ ∀ 𝑦𝐾 ( ( 𝐹𝑃 ) ∈ 𝑦 → ∃ 𝑥𝐽 ( 𝑃𝑥 ∧ ( 𝐹𝑥 ) ⊆ 𝑦 ) ) ) ) )
4 3 simprbi ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → ( 𝐹 : 𝑋𝑌 ∧ ∀ 𝑦𝐾 ( ( 𝐹𝑃 ) ∈ 𝑦 → ∃ 𝑥𝐽 ( 𝑃𝑥 ∧ ( 𝐹𝑥 ) ⊆ 𝑦 ) ) ) )
5 4 simpld ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐹 : 𝑋𝑌 )