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Metamath Proof Explorer

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 ) )