Theorem elbasfv

Description: Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015)

	Ref	Expression
Hypotheses	elbasfv.s	|- S = ( F ` Z )
	elbasfv.b	|- B = ( Base ` S )
Assertion	elbasfv	|- ( X e. B -> Z e. _V )

Proof

Step	Hyp	Ref	Expression
1		elbasfv.s	|- S = ( F ` Z )
2		elbasfv.b	|- B = ( Base ` S )
3		n0i	|- ( X e. B -> -. B = (/) )
4		fvprc	|- ( -. Z e. _V -> ( F ` Z ) = (/) )
5	1 4	eqtrid	|- ( -. Z e. _V -> S = (/) )
6	5	fveq2d	|- ( -. Z e. _V -> ( Base ` S ) = ( Base ` (/) ) )
7		base0	|- (/) = ( Base ` (/) )
8	6 2 7	3eqtr4g	|- ( -. Z e. _V -> B = (/) )
9	3 8	nsyl2	|- ( X e. B -> Z e. _V )