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Theorem fnex 5593
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 5589. (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 5199 . . 3  |-  ( F  Fn  A  ->  Rel  F )
21adantr 453 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  Rel  F )
3 df-fn 4603 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
4 eleq1a 2322 . . . . . 6  |-  ( A  e.  B  ->  ( dom  F  =  A  ->  dom  F  e.  B ) )
54impcom 421 . . . . 5  |-  ( ( dom  F  =  A  /\  A  e.  B
)  ->  dom  F  e.  B )
6 resfunexg 5589 . . . . 5  |-  ( ( Fun  F  /\  dom  F  e.  B )  -> 
( F  |`  dom  F
)  e.  _V )
75, 6sylan2 462 . . . 4  |-  ( ( Fun  F  /\  ( dom  F  =  A  /\  A  e.  B )
)  ->  ( F  |` 
dom  F )  e. 
_V )
87anassrs 632 . . 3  |-  ( ( ( Fun  F  /\  dom  F  =  A )  /\  A  e.  B
)  ->  ( F  |` 
dom  F )  e. 
_V )
93, 8sylanb 460 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( F  |`  dom  F
)  e.  _V )
10 resdm 4900 . . . 4  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
1110eleq1d 2319 . . 3  |-  ( Rel 
F  ->  ( ( F  |`  dom  F )  e.  _V  <->  F  e.  _V ) )
1211biimpa 472 . 2  |-  ( ( Rel  F  /\  ( F  |`  dom  F )  e.  _V )  ->  F  e.  _V )
132, 9, 12syl2anc 645 1  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2727   dom cdm 4580    |` cres 4582   Rel wrel 4585   Fun wfun 4586    Fn wfn 4587
This theorem is referenced by:  funex  5595  fex  5601  offval  5937  ofrfval  5938  fndmeng  6822  cfsmolem  7780  axcc2lem  7946  prdsbas2  13242  prdsplusgval  13246  prdsmulrval  13248  prdsleval  13250  prdsdsval  13251  prdsvscaval  13252  brssc  13535  sscpwex  13536  ssclem  13540  isssc  13541  rescval2  13549  reschom  13551  rescabs  13554  isfuncd  13583  dprdw  15080  prdsmgp  15228  ptval  17097  elptr  17100  prdstopn  17154  qtoptop  17223  imastopn  17243  vdgrfval  23060  trpredex  23408  wfrlem15  23438  cur1vald  24365  domrancur1clem  24367  domrancur1c  24368  valcurfn1  24370  intopcoaconb  24706  intopcoaconc  24707  dsmmbas2  26369  dsmmelbas  26371
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608
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