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Theorem elfm 20426
Description: An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
elfm  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( A  e.  ( ( X  FilMap  F ) `
 B )  <->  ( A  C_  X  /\  E. x  e.  B  ( F " x )  C_  A
) ) )
Distinct variable groups:    x, B    x, C    x, F    x, X    x, A    x, Y

Proof of Theorem elfm
Dummy variables  t 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmval 20422 . . 3  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( X filGen ran  ( t  e.  B  |->  ( F " t
) ) ) )
21eleq2d 2513 . 2  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( A  e.  ( ( X  FilMap  F ) `
 B )  <->  A  e.  ( X filGen ran  ( t  e.  B  |->  ( F
" t ) ) ) ) )
3 eqid 2443 . . . . 5  |-  ran  (
t  e.  B  |->  ( F " t ) )  =  ran  (
t  e.  B  |->  ( F " t ) )
43fbasrn 20363 . . . 4  |-  ( ( B  e.  ( fBas `  Y )  /\  F : Y --> X  /\  X  e.  C )  ->  ran  ( t  e.  B  |->  ( F " t
) )  e.  (
fBas `  X )
)
543comr 1205 . . 3  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  ran  ( t  e.  B  |->  ( F " t
) )  e.  (
fBas `  X )
)
6 elfg 20350 . . 3  |-  ( ran  ( t  e.  B  |->  ( F " t
) )  e.  (
fBas `  X )  ->  ( A  e.  ( X filGen ran  ( t  e.  B  |->  ( F
" t ) ) )  <->  ( A  C_  X  /\  E. y  e. 
ran  ( t  e.  B  |->  ( F "
t ) ) y 
C_  A ) ) )
75, 6syl 16 . 2  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( A  e.  ( X filGen ran  ( t  e.  B  |->  ( F
" t ) ) )  <->  ( A  C_  X  /\  E. y  e. 
ran  ( t  e.  B  |->  ( F "
t ) ) y 
C_  A ) ) )
8 simpr 461 . . . . . 6  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  x  e.  B )
9 eqid 2443 . . . . . 6  |-  ( F
" x )  =  ( F " x
)
10 imaeq2 5323 . . . . . . . 8  |-  ( t  =  x  ->  ( F " t )  =  ( F " x
) )
1110eqeq2d 2457 . . . . . . 7  |-  ( t  =  x  ->  (
( F " x
)  =  ( F
" t )  <->  ( F " x )  =  ( F " x ) ) )
1211rspcev 3196 . . . . . 6  |-  ( ( x  e.  B  /\  ( F " x )  =  ( F "
x ) )  ->  E. t  e.  B  ( F " x )  =  ( F "
t ) )
138, 9, 12sylancl 662 . . . . 5  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  E. t  e.  B  ( F " x )  =  ( F " t ) )
14 simpl1 1000 . . . . . . 7  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  X  e.  C )
15 imassrn 5338 . . . . . . . 8  |-  ( F
" x )  C_  ran  F
16 frn 5727 . . . . . . . . . 10  |-  ( F : Y --> X  ->  ran  F  C_  X )
17163ad2ant3 1020 . . . . . . . . 9  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  ran  F  C_  X )
1817adantr 465 . . . . . . . 8  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ran  F  C_  X )
1915, 18syl5ss 3500 . . . . . . 7  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( F " x )  C_  X
)
2014, 19ssexd 4584 . . . . . 6  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( F " x )  e.  _V )
21 eqid 2443 . . . . . . 7  |-  ( t  e.  B  |->  ( F
" t ) )  =  ( t  e.  B  |->  ( F "
t ) )
2221elrnmpt 5239 . . . . . 6  |-  ( ( F " x )  e.  _V  ->  (
( F " x
)  e.  ran  (
t  e.  B  |->  ( F " t ) )  <->  E. t  e.  B  ( F " x )  =  ( F "
t ) ) )
2320, 22syl 16 . . . . 5  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( ( F " x )  e. 
ran  ( t  e.  B  |->  ( F "
t ) )  <->  E. t  e.  B  ( F " x )  =  ( F " t ) ) )
2413, 23mpbird 232 . . . 4  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( F " x )  e.  ran  ( t  e.  B  |->  ( F " t
) ) )
2510cbvmptv 4528 . . . . . . 7  |-  ( t  e.  B  |->  ( F
" t ) )  =  ( x  e.  B  |->  ( F "
x ) )
2625elrnmpt 5239 . . . . . 6  |-  ( y  e.  ran  ( t  e.  B  |->  ( F
" t ) )  ->  ( y  e. 
ran  ( t  e.  B  |->  ( F "
t ) )  <->  E. x  e.  B  y  =  ( F " x ) ) )
2726ibi 241 . . . . 5  |-  ( y  e.  ran  ( t  e.  B  |->  ( F
" t ) )  ->  E. x  e.  B  y  =  ( F " x ) )
2827adantl 466 . . . 4  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  y  e.  ran  ( t  e.  B  |->  ( F " t
) ) )  ->  E. x  e.  B  y  =  ( F " x ) )
29 simpr 461 . . . . 5  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  y  =  ( F " x ) )  ->  y  =  ( F " x ) )
3029sseq1d 3516 . . . 4  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  y  =  ( F " x ) )  ->  ( y  C_  A  <->  ( F "
x )  C_  A
) )
3124, 28, 30rexxfrd 4652 . . 3  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( E. y  e. 
ran  ( t  e.  B  |->  ( F "
t ) ) y 
C_  A  <->  E. x  e.  B  ( F " x )  C_  A
) )
3231anbi2d 703 . 2  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( A  C_  X  /\  E. y  e. 
ran  ( t  e.  B  |->  ( F "
t ) ) y 
C_  A )  <->  ( A  C_  X  /\  E. x  e.  B  ( F " x )  C_  A
) ) )
332, 7, 323bitrd 279 1  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( A  e.  ( ( X  FilMap  F ) `
 B )  <->  ( A  C_  X  /\  E. x  e.  B  ( F " x )  C_  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   E.wrex 2794   _Vcvv 3095    C_ wss 3461    |-> cmpt 4495   ran crn 4990   "cima 4992   -->wf 5574   ` cfv 5578  (class class class)co 6281   fBascfbas 18385   filGencfg 18386    FilMap cfm 20412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-fbas 18395  df-fg 18396  df-fm 20417
This theorem is referenced by:  elfm2  20427  fmfg  20428  rnelfm  20432  fmfnfmlem1  20433  fmfnfm  20437  fmco  20440  flfnei  20470  isflf  20472  isfcf  20513  filnetlem4  30175
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