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Theorem fmfg 17982
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.)
Hypothesis
Ref Expression
elfm2.l  |-  L  =  ( Y filGen B )
Assertion
Ref Expression
fmfg  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( ( X 
FilMap  F ) `  L
) )

Proof of Theorem fmfg
Dummy variables  s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfm2.l . . . 4  |-  L  =  ( Y filGen B )
21elfm2 17981 . . 3  |-  ( ( 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 17911 . . . . . 6  |-  ( B  e.  ( fBas `  Y
)  ->  ( Y filGen B )  e.  ( Fil `  Y ) )
41, 3syl5eqel 2521 . . . . 5  |-  ( B  e.  ( fBas `  Y
)  ->  L  e.  ( Fil `  Y ) )
5 filfbas 17881 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  ( fBas `  Y )
)
64, 5syl 16 . . . 4  |-  ( B  e.  ( fBas `  Y
)  ->  L  e.  ( fBas `  Y )
)
7 elfm 17980 . . . 4  |-  ( ( 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
) ) )
86, 7syl3an2 1219 . . 3  |-  ( ( 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
) ) )
92, 8bitr4d 249 . 2  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( x  e.  ( ( X  FilMap  F ) `
 B )  <->  x  e.  ( ( X  FilMap  F ) `  L ) ) )
109eqrdv 2435 1  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( ( X 
FilMap  F ) `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2707    C_ wss 3321   "cima 4882   -->wf 5451   ` cfv 5455  (class class class)co 6082   fBascfbas 16690   filGencfg 16691   Filcfil 17878    FilMap cfm 17966
This theorem is referenced by:  fmfnfm  17991  cmetcaulem  19242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-fbas 16700  df-fg 16701  df-fil 17879  df-fm 17971
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