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Theorem fnex 5757
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 5753. See fnexALT 5758 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 5358 . . 3  |-  ( F  Fn  A  ->  Rel  F )
21adantr 451 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  Rel  F )
3 df-fn 5274 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
4 eleq1a 2365 . . . . . 6  |-  ( A  e.  B  ->  ( dom  F  =  A  ->  dom  F  e.  B ) )
54impcom 419 . . . . 5  |-  ( ( dom  F  =  A  /\  A  e.  B
)  ->  dom  F  e.  B )
6 resfunexg 5753 . . . . 5  |-  ( ( Fun  F  /\  dom  F  e.  B )  -> 
( F  |`  dom  F
)  e.  _V )
75, 6sylan2 460 . . . 4  |-  ( ( Fun  F  /\  ( dom  F  =  A  /\  A  e.  B )
)  ->  ( F  |` 
dom  F )  e. 
_V )
87anassrs 629 . . 3  |-  ( ( ( Fun  F  /\  dom  F  =  A )  /\  A  e.  B
)  ->  ( F  |` 
dom  F )  e. 
_V )
93, 8sylanb 458 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( F  |`  dom  F
)  e.  _V )
10 resdm 5009 . . . 4  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
1110eleq1d 2362 . . 3  |-  ( Rel 
F  ->  ( ( F  |`  dom  F )  e.  _V  <->  F  e.  _V ) )
1211biimpa 470 . 2  |-  ( ( Rel  F  /\  ( F  |`  dom  F )  e.  _V )  ->  F  e.  _V )
132, 9, 12syl2anc 642 1  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   dom cdm 4705    |` cres 4707   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266
This theorem is referenced by:  funex  5759  fex  5765  offval  6101  ofrfval  6102  fndmeng  6953  cfsmolem  7912  axcc2lem  8078  unirnfdomd  8205  prdsbas2  13384  prdsplusgval  13388  prdsmulrval  13390  prdsleval  13392  prdsdsval  13393  prdsvscaval  13394  brssc  13707  sscpwex  13708  ssclem  13712  isssc  13713  rescval2  13721  reschom  13723  rescabs  13726  isfuncd  13755  dprdw  15261  prdsmgp  15409  ptval  17281  elptr  17284  prdstopn  17338  qtoptop  17407  imastopn  17427  ofcfval  23474  vdgrfval  23904  trpredex  24311  wfrlem15  24341  cur1vald  25302  domrancur1clem  25304  domrancur1c  25305  valcurfn1  25307  intopcoaconb  25643  intopcoaconc  25644  dsmmbas2  27306  dsmmelbas  27308  stoweidlem27  27879  stoweidlem59  27911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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