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Theorem elfm 17642
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 17638 . . 3  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( X filGen ran  ( t  e.  B  |->  ( F " t
) ) ) )
21eleq2d 2350 . 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 2283 . . . . 5  |-  ran  (
t  e.  B  |->  ( F " t ) )  =  ran  (
t  e.  B  |->  ( F " t ) )
43fbasrn 17579 . . . 4  |-  ( ( B  e.  ( fBas `  Y )  /\  F : Y --> X  /\  X  e.  C )  ->  ran  ( t  e.  B  |->  ( F " t
) )  e.  (
fBas `  X )
)
543comr 1159 . . 3  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  ran  ( t  e.  B  |->  ( F " t
) )  e.  (
fBas `  X )
)
6 elfg 17566 . . 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 15 . 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 447 . . . . . 6  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  x  e.  B )
9 eqid 2283 . . . . . 6  |-  ( F
" x )  =  ( F " x
)
10 imaeq2 5008 . . . . . . . 8  |-  ( t  =  x  ->  ( F " t )  =  ( F " x
) )
1110eqeq2d 2294 . . . . . . 7  |-  ( t  =  x  ->  (
( F " x
)  =  ( F
" t )  <->  ( F " x )  =  ( F " x ) ) )
1211rspcev 2884 . . . . . 6  |-  ( ( x  e.  B  /\  ( F " x )  =  ( F "
x ) )  ->  E. t  e.  B  ( F " x )  =  ( F "
t ) )
138, 9, 12sylancl 643 . . . . 5  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  E. t  e.  B  ( F " x )  =  ( F " t ) )
14 imassrn 5025 . . . . . . . 8  |-  ( F
" x )  C_  ran  F
15 frn 5395 . . . . . . . . . 10  |-  ( F : Y --> X  ->  ran  F  C_  X )
16153ad2ant3 978 . . . . . . . . 9  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  ran  F  C_  X )
1716adantr 451 . . . . . . . 8  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ran  F  C_  X )
1814, 17syl5ss 3190 . . . . . . 7  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( F " x )  C_  X
)
19 simpl1 958 . . . . . . 7  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  X  e.  C )
20 ssexg 4160 . . . . . . 7  |-  ( ( ( F " x
)  C_  X  /\  X  e.  C )  ->  ( F " x
)  e.  _V )
2118, 19, 20syl2anc 642 . . . . . 6  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( F " x )  e.  _V )
22 eqid 2283 . . . . . . 7  |-  ( t  e.  B  |->  ( F
" t ) )  =  ( t  e.  B  |->  ( F "
t ) )
2322elrnmpt 4926 . . . . . 6  |-  ( ( F " x )  e.  _V  ->  (
( F " x
)  e.  ran  (
t  e.  B  |->  ( F " t ) )  <->  E. t  e.  B  ( F " x )  =  ( F "
t ) ) )
2421, 23syl 15 . . . . 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 ) ) )
2513, 24mpbird 223 . . . 4  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( F " x )  e.  ran  ( t  e.  B  |->  ( F " t
) ) )
2610cbvmptv 4111 . . . . . . 7  |-  ( t  e.  B  |->  ( F
" t ) )  =  ( x  e.  B  |->  ( F "
x ) )
2726elrnmpt 4926 . . . . . 6  |-  ( y  e.  ran  ( t  e.  B  |->  ( F
" t ) )  ->  ( y  e. 
ran  ( t  e.  B  |->  ( F "
t ) )  <->  E. x  e.  B  y  =  ( F " x ) ) )
2827ibi 232 . . . . 5  |-  ( y  e.  ran  ( t  e.  B  |->  ( F
" t ) )  ->  E. x  e.  B  y  =  ( F " x ) )
2928adantl 452 . . . 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 ) )
30 simpr 447 . . . . 5  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  y  =  ( F " x ) )  ->  y  =  ( F " x ) )
3130sseq1d 3205 . . . 4  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  y  =  ( F " x ) )  ->  ( y  C_  A  <->  ( F "
x )  C_  A
) )
3225, 29, 31rexxfrd 4549 . . 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
) )
3332anbi2d 684 . 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
) ) )
342, 7, 333bitrd 270 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152    e. cmpt 4077   ran crn 4690   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   fBascfbas 17518   filGencfg 17519    FilMap cfm 17628
This theorem is referenced by:  elfm2  17643  fmfg  17644  rnelfm  17648  fmfnfmlem1  17649  fmfnfm  17653  fmco  17656  flfnei  17686  isflf  17688  isfcf  17729  filnetlem4  26330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-fbas 17520  df-fg 17521  df-fm 17633
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