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Theorem fnex 5702
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 5698. (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 5307 . . 3  |-  ( F  Fn  A  ->  Rel  F )
21adantr 453 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  Rel  F )
3 df-fn 5224 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
4 eleq1a 2353 . . . . . 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 5698 . . . . 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 631 . . 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 4992 . . . 4  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
1110eleq1d 2350 . . 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 644 1  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   _Vcvv 2789   dom cdm 4688    |` cres 4690   Rel wrel 4693   Fun wfun 5215    Fn wfn 5216
This theorem is referenced by:  funex  5704  fex  5710  offval  6046  ofrfval  6047  fndmeng  6932  cfsmolem  7891  axcc2lem  8057  unirnfdomd  8184  prdsbas2  13362  prdsplusgval  13366  prdsmulrval  13368  prdsleval  13370  prdsdsval  13371  prdsvscaval  13372  brssc  13685  sscpwex  13686  ssclem  13690  isssc  13691  rescval2  13699  reschom  13701  rescabs  13704  isfuncd  13733  dprdw  15239  prdsmgp  15387  ptval  17259  elptr  17262  prdstopn  17316  qtoptop  17385  imastopn  17405  vdgrfval  23293  trpredex  23641  wfrlem15  23671  cur1vald  24598  domrancur1clem  24600  domrancur1c  24601  valcurfn1  24603  intopcoaconb  24939  intopcoaconc  24940  dsmmbas2  26602  dsmmelbas  26604  stoweidlem27  27175  stoweidlem59  27207
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229
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