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Theorem fnex 5920
Description: If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5916. See fnexALT 5921 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fnex  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )

Proof of Theorem fnex
StepHypRef Expression
1 fnrel 5502 . . 3  |-  ( F  Fn  A  ->  Rel  F )
21adantr 452 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  Rel  F )
3 df-fn 5416 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
4 eleq1a 2473 . . . . . 6  |-  ( A  e.  B  ->  ( dom  F  =  A  ->  dom  F  e.  B ) )
54impcom 420 . . . . 5  |-  ( ( dom  F  =  A  /\  A  e.  B
)  ->  dom  F  e.  B )
6 resfunexg 5916 . . . . 5  |-  ( ( Fun  F  /\  dom  F  e.  B )  -> 
( F  |`  dom  F
)  e.  _V )
75, 6sylan2 461 . . . 4  |-  ( ( Fun  F  /\  ( dom  F  =  A  /\  A  e.  B )
)  ->  ( F  |` 
dom  F )  e. 
_V )
87anassrs 630 . . 3  |-  ( ( ( Fun  F  /\  dom  F  =  A )  /\  A  e.  B
)  ->  ( F  |` 
dom  F )  e. 
_V )
93, 8sylanb 459 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( F  |`  dom  F
)  e.  _V )
10 resdm 5143 . . . 4  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
1110eleq1d 2470 . . 3  |-  ( Rel 
F  ->  ( ( F  |`  dom  F )  e.  _V  <->  F  e.  _V ) )
1211biimpa 471 . 2  |-  ( ( Rel  F  /\  ( F  |`  dom  F )  e.  _V )  ->  F  e.  _V )
132, 9, 12syl2anc 643 1  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   dom cdm 4837    |` cres 4839   Rel wrel 4842   Fun wfun 5407    Fn wfn 5408
This theorem is referenced by:  funex  5922  fex  5928  offval  6271  ofrfval  6272  fndmeng  7142  cfsmolem  8106  axcc2lem  8272  unirnfdomd  8398  prdsbas2  13646  prdsplusgval  13650  prdsmulrval  13652  prdsleval  13654  prdsdsval  13655  prdsvscaval  13656  brssc  13969  sscpwex  13970  ssclem  13974  isssc  13975  rescval2  13983  reschom  13985  rescabs  13988  isfuncd  14017  dprdw  15523  prdsmgp  15671  ptval  17555  elptr  17558  prdstopn  17613  qtoptop  17685  imastopn  17705  vdgrfval  21619  ofcfval  24434  dya2iocuni  24586  trpredex  25454  wfrlem15  25484  dsmmbas2  27071  dsmmelbas  27073  stoweidlem27  27643  stoweidlem59  27675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421
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