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Theorem funex 5677
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 5675. (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 5222 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fnex 5675 . 2  |-  ( ( F  Fn  dom  F  /\  dom  F  e.  B
)  ->  F  e.  _V )
31, 2sylanb 460 1  |-  ( ( Fun  F  /\  dom  F  e.  B )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1621   _Vcvv 2763   dom cdm 4661   Fun wfun 4667    Fn wfn 4668
This theorem is referenced by:  opabex  5678  mptexg  5679  funrnex  5681  oprabexd  5894  oprabex  5895  mpt2exxg  6129  tfrlem14  6375  hartogslem2  7226  harwdom  7272  trset  24760  ltrset  24770  rltrset  24781
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689
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