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Theorem fmfg 19527
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 19526 . . 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 19456 . . . . . 6  |-  ( B  e.  ( fBas `  Y
)  ->  ( Y filGen B )  e.  ( Fil `  Y ) )
41, 3syl5eqel 2527 . . . . 5  |-  ( B  e.  ( fBas `  Y
)  ->  L  e.  ( Fil `  Y ) )
5 filfbas 19426 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  ( fBas `  Y )
)
64, 5syl 16 . . . 4  |-  ( B  e.  ( fBas `  Y
)  ->  L  e.  ( fBas `  Y )
)
7 elfm 19525 . . . 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 1252 . . 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 256 . 2  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( x  e.  ( ( X  FilMap  F ) `
 B )  <->  x  e.  ( ( X  FilMap  F ) `  L ) ) )
109eqrdv 2441 1  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( ( X 
FilMap  F ) `  L
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2721    C_ wss 3333   "cima 4848   -->wf 5419   ` cfv 5423  (class class class)co 6096   fBascfbas 17809   filGencfg 17810   Filcfil 19423    FilMap cfm 19511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-fbas 17819  df-fg 17820  df-fil 19424  df-fm 19516
This theorem is referenced by:  fmfnfm  19536  cmetcaulem  20804
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