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Theorem fmfg 17660
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 17659 . . 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 17589 . . . . . 6  |-  ( B  e.  ( fBas `  Y
)  ->  ( Y filGen B )  e.  ( Fil `  Y ) )
41, 3syl5eqel 2380 . . . . 5  |-  ( B  e.  ( fBas `  Y
)  ->  L  e.  ( Fil `  Y ) )
5 filfbas 17559 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  ( fBas `  Y )
)
64, 5syl 15 . . . 4  |-  ( B  e.  ( fBas `  Y
)  ->  L  e.  ( fBas `  Y )
)
7 elfm 17658 . . . 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 1216 . . 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 247 . 2  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( x  e.  ( ( X  FilMap  F ) `
 B )  <->  x  e.  ( ( X  FilMap  F ) `  L ) ) )
109eqrdv 2294 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   fBascfbas 17534   filGencfg 17535   Filcfil 17556    FilMap cfm 17644
This theorem is referenced by:  fmfnfm  17669  cmetcaulem  18730  cnpflf4  25667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649
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