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Theorem elfm 20211
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 20207 . . 3  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  B )  =  ( X filGen ran  ( t  e.  B  |->  ( F " t
) ) ) )
21eleq2d 2537 . 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 2467 . . . . 5  |-  ran  (
t  e.  B  |->  ( F " t ) )  =  ran  (
t  e.  B  |->  ( F " t ) )
43fbasrn 20148 . . . 4  |-  ( ( B  e.  ( fBas `  Y )  /\  F : Y --> X  /\  X  e.  C )  ->  ran  ( t  e.  B  |->  ( F " t
) )  e.  (
fBas `  X )
)
543comr 1204 . . 3  |-  ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  ran  ( t  e.  B  |->  ( F " t
) )  e.  (
fBas `  X )
)
6 elfg 20135 . . 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 2467 . . . . . 6  |-  ( F
" x )  =  ( F " x
)
10 imaeq2 5333 . . . . . . . 8  |-  ( t  =  x  ->  ( F " t )  =  ( F " x
) )
1110eqeq2d 2481 . . . . . . 7  |-  ( t  =  x  ->  (
( F " x
)  =  ( F
" t )  <->  ( F " x )  =  ( F " x ) ) )
1211rspcev 3214 . . . . . 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 999 . . . . . . 7  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  X  e.  C )
15 imassrn 5348 . . . . . . . 8  |-  ( F
" x )  C_  ran  F
16 frn 5737 . . . . . . . . . 10  |-  ( F : Y --> X  ->  ran  F  C_  X )
17163ad2ant3 1019 . . . . . . . . 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 3515 . . . . . . 7  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( F " x )  C_  X
)
2014, 19ssexd 4594 . . . . . 6  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  x  e.  B
)  ->  ( F " x )  e.  _V )
21 eqid 2467 . . . . . . 7  |-  ( t  e.  B  |->  ( F
" t ) )  =  ( t  e.  B  |->  ( F "
t ) )
2221elrnmpt 5249 . . . . . 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 4538 . . . . . . 7  |-  ( t  e.  B  |->  ( F
" t ) )  =  ( x  e.  B  |->  ( F "
x ) )
2625elrnmpt 5249 . . . . . 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 3531 . . . 4  |-  ( ( ( X  e.  C  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  /\  y  =  ( F " x ) )  ->  ( y  C_  A  <->  ( F "
x )  C_  A
) )
3124, 28, 30rexxfrd 4662 . . 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 973    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113    C_ wss 3476    |-> cmpt 4505   ran crn 5000   "cima 5002   -->wf 5584   ` cfv 5588  (class class class)co 6284   fBascfbas 18205   filGencfg 18206    FilMap cfm 20197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-fbas 18215  df-fg 18216  df-fm 20202
This theorem is referenced by:  elfm2  20212  fmfg  20213  rnelfm  20217  fmfnfmlem1  20218  fmfnfm  20222  fmco  20225  flfnei  20255  isflf  20257  isfcf  20298  filnetlem4  29830
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