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 = U. J
	iscnp2.2	|- Y = U. K
Assertion	cnpf	|- ( F e. ( ( J CnP K ) ` P ) -> F : X --> Y )

Proof

Step	Hyp	Ref	Expression
1		iscnp2.1	|- X = U. J
2		iscnp2.2	|- Y = U. K
3	1 2	iscnp2	|- ( F e. ( ( J CnP K ) ` P ) <-> ( ( J e. Top /\ K e. Top /\ P e. X ) /\ ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) ) )
4	3	simprbi	|- ( F e. ( ( J CnP K ) ` P ) -> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) )
5	4	simpld	|- ( F e. ( ( J CnP K ) ` P ) -> F : X --> Y )