Theorem rhmcl

Description: Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011) (Revised by AV, 10-Jan-2025)

	Ref	Expression
Hypotheses	rhmf.b	|- B = ( Base ` R )
	rhmf.c	|- C = ( Base ` S )
Assertion	rhmcl	|- ( ( ( R e. Ring /\ S e. Ring /\ F e. ( R RingHom S ) ) /\ A e. B ) -> ( F ` A ) e. C )

Proof

Step	Hyp	Ref	Expression
1		rhmf.b	|- B = ( Base ` R )
2		rhmf.c	|- C = ( Base ` S )
3	1 2	rhmf	|- ( F e. ( R RingHom S ) -> F : B --> C )
4	3	3ad2ant3	|- ( ( R e. Ring /\ S e. Ring /\ F e. ( R RingHom S ) ) -> F : B --> C )
5	4	ffvelcdmda	|- ( ( ( R e. Ring /\ S e. Ring /\ F e. ( R RingHom S ) ) /\ A e. B ) -> ( F ` A ) e. C )