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Metamath Proof Explorer

Theorem cnpcl

Description: The value of a continuous function from J to K at point P belongs to the underlying set of topology K . (Contributed by FL, 27-Dec-2006) (Revised by Mario Carneiro, 21-Aug-2015)

	Ref	Expression
Hypotheses	iscnp2.1	⊢ 𝑋 = ∪ 𝐽
	iscnp2.2	⊢ 𝑌 = ∪ 𝐾
Assertion	cnpcl	⊢ ( ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		iscnp2.1	⊢ 𝑋 = ∪ 𝐽
2		iscnp2.2	⊢ 𝑌 = ∪ 𝐾
3	1 2	cnpf	⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐹 : 𝑋 ⟶ 𝑌 )
4	3	ffvelcdmda	⊢ ( ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑌 )