Theorem fvmptelrn

Description: The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypothesis	fvmptelrn.1	|- ( ph -> ( x e. A |-> B ) : A --> C )
Assertion	fvmptelrn	|- ( ( ph /\ x e. A ) -> B e. C )

Proof

Step	Hyp	Ref	Expression
1		fvmptelrn.1	|- ( ph -> ( x e. A |-> B ) : A --> C )
2		eqid	|- ( x e. A |-> B ) = ( x e. A |-> B )
3	2	fmpt	|- ( A. x e. A B e. C <-> ( x e. A |-> B ) : A --> C )
4	1 3	sylibr	|- ( ph -> A. x e. A B e. C )
5	4	r19.21bi	|- ( ( ph /\ x e. A ) -> B e. C )