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Theorem fnex 5759
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 5758. (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 5333 . . 3  |-  ( F  Fn  A  ->  Rel  F )
21adantr 276 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  Rel  F )
3 df-fn 5238 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
4 eleq1a 2261 . . . . . 6  |-  ( A  e.  B  ->  ( dom  F  =  A  ->  dom  F  e.  B ) )
54impcom 125 . . . . 5  |-  ( ( dom  F  =  A  /\  A  e.  B
)  ->  dom  F  e.  B )
6 resfunexg 5758 . . . . 5  |-  ( ( Fun  F  /\  dom  F  e.  B )  -> 
( F  |`  dom  F
)  e.  _V )
75, 6sylan2 286 . . . 4  |-  ( ( Fun  F  /\  ( dom  F  =  A  /\  A  e.  B )
)  ->  ( F  |` 
dom  F )  e. 
_V )
87anassrs 400 . . 3  |-  ( ( ( Fun  F  /\  dom  F  =  A )  /\  A  e.  B
)  ->  ( F  |` 
dom  F )  e. 
_V )
93, 8sylanb 284 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( F  |`  dom  F
)  e.  _V )
10 resdm 4964 . . . 4  |-  ( Rel 
F  ->  ( F  |` 
dom  F )  =  F )
1110eleq1d 2258 . . 3  |-  ( Rel 
F  ->  ( ( F  |`  dom  F )  e.  _V  <->  F  e.  _V ) )
1211biimpa 296 . 2  |-  ( ( Rel  F  /\  ( F  |`  dom  F )  e.  _V )  ->  F  e.  _V )
132, 9, 12syl2anc 411 1  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752   dom cdm 4644    |` cres 4646   Rel wrel 4649   Fun wfun 5229    Fn wfn 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243
This theorem is referenced by:  funex  5760  fex  5766  offval  6115  ofrfval  6116  tfrlemibex  6355  tfr1onlembex  6371  fndmeng  6837  cc2lem  7296  frecfzennn  10459  xpscf  12826  mulgval  13079  mulgfng  13081  invrfvald  13489
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