Theorem elpreimad

Description: Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	elpreimad.f	|- ( ph -> F Fn A )
	elpreimad.b	|- ( ph -> B e. A )
	elpreimad.c	|- ( ph -> ( F ` B ) e. C )
Assertion	elpreimad	|- ( ph -> B e. ( `' F " C ) )

Proof

Step	Hyp	Ref	Expression
1		elpreimad.f	|- ( ph -> F Fn A )
2		elpreimad.b	|- ( ph -> B e. A )
3		elpreimad.c	|- ( ph -> ( F ` B ) e. C )
4		elpreima	|- ( F Fn A -> ( B e. ( `' F " C ) <-> ( B e. A /\ ( F ` B ) e. C ) ) )
5	1 4	syl	|- ( ph -> ( B e. ( `' F " C ) <-> ( B e. A /\ ( F ` B ) e. C ) ) )
6	2 3 5	mpbir2and	|- ( ph -> B e. ( `' F " C ) )