Theorem imaelfm

Description: An image of a filter element is in the image filter. (Contributed by Jeff Hankins, 5-Oct-2009) (Revised by Stefan O'Rear, 6-Aug-2015)

	Ref	Expression
Hypothesis	imaelfm.l	|- L = ( Y filGen B )
Assertion	imaelfm	|- ( ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ S e. L ) -> ( F " S ) e. ( ( X FilMap F ) ` B ) )

Proof

Step	Hyp	Ref	Expression
1		imaelfm.l	|- L = ( Y filGen B )
2		fimass	|- ( F : Y --> X -> ( F " S ) C_ X )
3	2	3ad2ant3	|- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( F " S ) C_ X )
4		ssid	|- ( F " S ) C_ ( F " S )
5		imaeq2	|- ( x = S -> ( F " x ) = ( F " S ) )
6	5	sseq1d	|- ( x = S -> ( ( F " x ) C_ ( F " S ) <-> ( F " S ) C_ ( F " S ) ) )
7	6	rspcev	|- ( ( S e. L /\ ( F " S ) C_ ( F " S ) ) -> E. x e. L ( F " x ) C_ ( F " S ) )
8	4 7	mpan2	|- ( S e. L -> E. x e. L ( F " x ) C_ ( F " S ) )
9	3 8	anim12i	|- ( ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ S e. L ) -> ( ( F " S ) C_ X /\ E. x e. L ( F " x ) C_ ( F " S ) ) )
10	1	elfm2	|- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( F " S ) e. ( ( X FilMap F ) ` B ) <-> ( ( F " S ) C_ X /\ E. x e. L ( F " x ) C_ ( F " S ) ) ) )
11	10	adantr	|- ( ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ S e. L ) -> ( ( F " S ) e. ( ( X FilMap F ) ` B ) <-> ( ( F " S ) C_ X /\ E. x e. L ( F " x ) C_ ( F " S ) ) ) )
12	9 11	mpbird	|- ( ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ S e. L ) -> ( F " S ) e. ( ( X FilMap F ) ` B ) )