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Theorem elfm 19532
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 19528 . . 3  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( X filGen ran  ( t  e.  B  |->  ( F " t
) ) ) )
21eleq2d 2510 . 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 19469 . . . 4  |-  ( ( B  e.  ( fBas `  Y )  /\  F : Y --> X  /\  X  e.  C )  ->  ran  ( t  e.  B  |->  ( F " t
) )  e.  (
fBas `  X )
)
543comr 1195 . . 3  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  ran  ( t  e.  B  |->  ( F " t
) )  e.  (
fBas `  X )
)
6 elfg 19456 . . 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 5177 . . . . . . . 8  |-  ( t  =  x  ->  ( F " t )  =  ( F " x
) )
1110eqeq2d 2454 . . . . . . 7  |-  ( t  =  x  ->  (
( F " x
)  =  ( F
" t )  <->  ( F " x )  =  ( F " x ) ) )
1211rspcev 3085 . . . . . 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 991 . . . . . . 7  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  X  e.  C )
15 imassrn 5192 . . . . . . . 8  |-  ( F
" x )  C_  ran  F
16 frn 5577 . . . . . . . . . 10  |-  ( F : Y --> X  ->  ran  F  C_  X )
17163ad2ant3 1011 . . . . . . . . 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 3379 . . . . . . 7  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( F " x )  C_  X
)
2014, 19ssexd 4451 . . . . . 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 5098 . . . . . 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 4395 . . . . . . 7  |-  ( t  e.  B  |->  ( F
" t ) )  =  ( x  e.  B  |->  ( F "
x ) )
2625elrnmpt 5098 . . . . . 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 3395 . . . 4  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  y  =  ( F " x ) )  ->  ( y  C_  A  <->  ( F "
x )  C_  A
) )
3124, 28, 30rexxfrd 4519 . . 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 965    = wceq 1369    e. wcel 1756   E.wrex 2728   _Vcvv 2984    C_ wss 3340    e. cmpt 4362   ran crn 4853   "cima 4855   -->wf 5426   ` cfv 5430  (class class class)co 6103   fBascfbas 17816   filGencfg 17817    FilMap cfm 19518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-fbas 17826  df-fg 17827  df-fm 19523
This theorem is referenced by:  elfm2  19533  fmfg  19534  rnelfm  19538  fmfnfmlem1  19539  fmfnfm  19543  fmco  19546  flfnei  19576  isflf  19578  isfcf  19619  filnetlem4  28614
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