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Theorem fmfg 20888
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 20887 . . 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 20817 . . . . . 6  |-  ( B  e.  ( fBas `  Y
)  ->  ( Y filGen B )  e.  ( Fil `  Y ) )
41, 3syl5eqel 2512 . . . . 5  |-  ( B  e.  ( fBas `  Y
)  ->  L  e.  ( Fil `  Y ) )
5 filfbas 20787 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  ( fBas `  Y )
)
64, 5syl 17 . . . 4  |-  ( B  e.  ( fBas `  Y
)  ->  L  e.  ( fBas `  Y )
)
7 elfm 20886 . . . 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 1298 . . 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 259 . 2  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( x  e.  ( ( X  FilMap  F ) `
 B )  <->  x  e.  ( ( X  FilMap  F ) `  L ) ) )
109eqrdv 2417 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   E.wrex 2774    C_ wss 3433   "cima 4848   -->wf 5588   ` cfv 5592  (class class class)co 6296   fBascfbas 18886   filGencfg 18887   Filcfil 20784    FilMap cfm 20872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  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 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-fbas 18895  df-fg 18896  df-fil 20785  df-fm 20877
This theorem is referenced by:  fmfnfm  20897  cmetcaulem  22164
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