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Theorem funrnex 5749
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 5745. (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 5745 . . 3  |-  ( ( Fun  F  /\  dom  F  e.  B )  ->  F  e.  _V )
21ex 423 . 2  |-  ( Fun 
F  ->  ( dom  F  e.  B  ->  F  e.  _V ) )
3 rnexg 4942 . 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 1686   _Vcvv 2790   dom cdm 4691   ran crn 4692   Fun wfun 5251
This theorem is referenced by:  zfrep6  5750  fornex  5752  tz7.48-3  6458  inf0  7324  noinfepOLD  7363  axcc2lem  8064  zorn2lem4  8128  fnct  23343  domrancur1b  25211  supnuf  25640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265
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