Theorem fvmptelcdmf

Description: The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 5-Jan-2025)

	Ref	Expression
Hypotheses	fvmptelcdmf.a	|- F/_ x A
	fvmptelcdmf.c	|- F/_ x C
	fvmptelcdmf.f	|- ( ph -> ( x e. A |-> B ) : A --> C )
Assertion	fvmptelcdmf	|- ( ( ph /\ x e. A ) -> B e. C )

Proof

Step	Hyp	Ref	Expression
1		fvmptelcdmf.a	|- F/_ x A
2		fvmptelcdmf.c	|- F/_ x C
3		fvmptelcdmf.f	|- ( ph -> ( x e. A |-> B ) : A --> C )
4		eqid	|- ( x e. A |-> B ) = ( x e. A |-> B )
5	1 2 4	fmptff	|- ( A. x e. A B e. C <-> ( x e. A |-> B ) : A --> C )
6	3 5	sylibr	|- ( ph -> A. x e. A B e. C )
7	6	r19.21bi	|- ( ( ph /\ x e. A ) -> B e. C )