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Theorem funrnex 5709
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 5705. (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 5705 . . 3  |-  ( ( Fun  F  /\  dom  F  e.  B )  ->  F  e.  _V )
21ex 423 . 2  |-  ( Fun 
F  ->  (  dom  F  e.  B  ->  F  e.  _V ) )
3 rnexg 4939 . 2  |-  ( F  e.  _V  ->  ran  F  e.  _V )
42, 3syl6com 31 1  |-  (  dom 
F  e.  B  -> 
( Fun  F  ->  ran 
F  e.  _V )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1685   _Vcvv 2789    dom cdm 4688   ran crn 4689   Fun wfun 5215
This theorem is referenced by:  zfrep6  5710  fornex  5712  tz7.48-3  6452  inf0  7318  noinfepOLD  7357  axcc2lem  8058  zorn2lem4  8122  domrancur1b  24611  supnuf  25040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229
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