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Theorem funex 5759
Description: If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5757. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
Assertion
Ref Expression
funex  |-  ( ( Fun  F  /\  dom  F  e.  B )  ->  F  e.  _V )

Proof of Theorem funex
StepHypRef Expression
1 funfn 5299 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fnex 5757 . 2  |-  ( ( F  Fn  dom  F  /\  dom  F  e.  B
)  ->  F  e.  _V )
31, 2sylanb 458 1  |-  ( ( Fun  F  /\  dom  F  e.  B )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   _Vcvv 2801   dom cdm 4705   Fun wfun 5265    Fn wfn 5266
This theorem is referenced by:  opabex  5760  mptexg  5761  funrnex  5763  oprabexd  5976  oprabex  5977  mpt2exxg  6211  tfrlem14  6423  hartogslem2  7274  harwdom  7320  abrexexd  23207  trset  25495  ltrset  25505  rltrset  25516
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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