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Metamath Proof Explorer

Theorem fmfg

Description: The image filter of a filter base is the same as the image filter of its generated filter. (Contributed by Jeff Hankins, 18-Nov-2009) (Revised by Stefan O'Rear, 6-Aug-2015)

Ref Expression
Hypothesis elfm2.l 
|- L = ( Y filGen B )
Assertion fmfg 
|- ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( X FilMap F ) ` B ) = ( ( X FilMap F ) ` L ) )

Proof

Step Hyp Ref Expression
1 elfm2.l 
 |-  L = ( Y filGen B )
2 1 elfm2 
 |-  ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( x e. ( ( X FilMap F ) ` B ) <-> ( x C_ X /\ E. s e. L ( F " s ) C_ x ) ) )
3 fgcl 
 |-  ( B e. ( fBas ` Y ) -> ( Y filGen B ) e. ( Fil ` Y ) )
4 1 3 eqeltrid 
 |-  ( B e. ( fBas ` Y ) -> L e. ( Fil ` Y ) )
5 filfbas 
 |-  ( L e. ( Fil ` Y ) -> L e. ( fBas ` Y ) )
6 4 5 syl 
 |-  ( B e. ( fBas ` Y ) -> L e. ( fBas ` Y ) )
7 elfm 
 |-  ( ( X e. C /\ L e. ( fBas ` Y ) /\ F : Y --> X ) -> ( x e. ( ( X FilMap F ) ` L ) <-> ( x C_ X /\ E. s e. L ( F " s ) C_ x ) ) )
8 6 7 syl3an2 
 |-  ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( x e. ( ( X FilMap F ) ` L ) <-> ( x C_ X /\ E. s e. L ( F " s ) C_ x ) ) )
9 2 8 bitr4d 
 |-  ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( x e. ( ( X FilMap F ) ` B ) <-> x e. ( ( X FilMap F ) ` L ) ) )
10 9 eqrdv 
 |-  ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( X FilMap F ) ` B ) = ( ( X FilMap F ) ` L ) )