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Theorem elfm 20740
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 20736 . . 3  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( X filGen ran  ( t  e.  B  |->  ( F " t
) ) ) )
21eleq2d 2472 . 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 2402 . . . . 5  |-  ran  (
t  e.  B  |->  ( F " t ) )  =  ran  (
t  e.  B  |->  ( F " t ) )
43fbasrn 20677 . . . 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 20664 . . 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 17 . 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 459 . . . . . 6  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  x  e.  B )
9 eqid 2402 . . . . . 6  |-  ( F
" x )  =  ( F " x
)
10 imaeq2 5153 . . . . . . . 8  |-  ( t  =  x  ->  ( F " t )  =  ( F " x
) )
1110eqeq2d 2416 . . . . . . 7  |-  ( t  =  x  ->  (
( F " x
)  =  ( F
" t )  <->  ( F " x )  =  ( F " x ) ) )
1211rspcev 3160 . . . . . 6  |-  ( ( x  e.  B  /\  ( F " x )  =  ( F "
x ) )  ->  E. t  e.  B  ( F " x )  =  ( F "
t ) )
138, 9, 12sylancl 660 . . . . 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 5168 . . . . . . . 8  |-  ( F
" x )  C_  ran  F
16 frn 5720 . . . . . . . . . 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 463 . . . . . . . 8  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ran  F  C_  X )
1915, 18syl5ss 3453 . . . . . . 7  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( F " x )  C_  X
)
2014, 19ssexd 4541 . . . . . 6  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( F " x )  e.  _V )
21 eqid 2402 . . . . . . 7  |-  ( t  e.  B  |->  ( F
" t ) )  =  ( t  e.  B  |->  ( F "
t ) )
2221elrnmpt 5070 . . . . . 6  |-  ( ( F " x )  e.  _V  ->  (
( F " x
)  e.  ran  (
t  e.  B  |->  ( F " t ) )  <->  E. t  e.  B  ( F " x )  =  ( F "
t ) ) )
2320, 22syl 17 . . . . 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 4487 . . . . . . 7  |-  ( t  e.  B  |->  ( F
" t ) )  =  ( x  e.  B  |->  ( F "
x ) )
2625elrnmpt 5070 . . . . . 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 464 . . . 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 459 . . . . 5  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  y  =  ( F " x ) )  ->  y  =  ( F " x ) )
3029sseq1d 3469 . . . 4  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  y  =  ( F " x ) )  ->  ( y  C_  A  <->  ( F "
x )  C_  A
) )
3124, 28, 30rexxfrd 4606 . . 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 702 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2755   _Vcvv 3059    C_ wss 3414    |-> cmpt 4453   ran crn 4824   "cima 4826   -->wf 5565   ` cfv 5569  (class class class)co 6278   fBascfbas 18726   filGencfg 18727    FilMap cfm 20726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-fbas 18736  df-fg 18737  df-fm 20731
This theorem is referenced by:  elfm2  20741  fmfg  20742  rnelfm  20746  fmfnfmlem1  20747  fmfnfm  20751  fmco  20754  flfnei  20784  isflf  20786  isfcf  20827  filnetlem4  30609
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