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Theorem funrnex 5926
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 5922. (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 5922 . . 3  |-  ( ( Fun  F  /\  dom  F  e.  B )  ->  F  e.  _V )
21ex 424 . 2  |-  ( Fun 
F  ->  ( dom  F  e.  B  ->  F  e.  _V ) )
3 rnexg 5090 . 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 4    e. wcel 1721   _Vcvv 2916   dom cdm 4837   ran crn 4838   Fun wfun 5407
This theorem is referenced by:  zfrep6  5927  fornex  5929  tz7.48-3  6660  inf0  7532  noinfepOLD  7571  axcc2lem  8272  zorn2lem4  8335  fnct  24058  hashimarn  27994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421
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