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Theorem funrnex 5599
Description: If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5595. (Contributed by NM, 11-Nov-1995.)
Assertion
Ref Expression
funrnex  |-  ( dom 
F  e.  B  -> 
( Fun  F  ->  ran 
F  e.  _V )
)

Proof of Theorem funrnex
StepHypRef Expression
1 funex 5595 . . 3  |-  ( ( Fun  F  /\  dom  F  e.  B )  ->  F  e.  _V )
21ex 425 . 2  |-  ( Fun 
F  ->  ( dom  F  e.  B  ->  F  e.  _V ) )
3 rnexg 4847 . 2  |-  ( F  e.  _V  ->  ran  F  e.  _V )
42, 3syl6com 33 1  |-  ( dom 
F  e.  B  -> 
( Fun  F  ->  ran 
F  e.  _V )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621   _Vcvv 2727   dom cdm 4580   ran crn 4581   Fun wfun 4586
This theorem is referenced by:  zfrep6  5600  fornex  5602  tz7.48-3  6342  inf0  7206  noinfepOLD  7245  axcc2lem  7946  zorn2lem4  8010  domrancur1b  24366  supnuf  24795
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608
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