MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funrnex Structured version   Unicode version

Theorem funrnex 6657
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 6057. (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 6057 . . 3  |-  ( ( Fun  F  /\  dom  F  e.  B )  ->  F  e.  _V )
21ex 434 . 2  |-  ( Fun 
F  ->  ( dom  F  e.  B  ->  F  e.  _V ) )
3 rnexg 6623 . 2  |-  ( F  e.  _V  ->  ran  F  e.  _V )
42, 3syl6com 35 1  |-  ( dom 
F  e.  B  -> 
( Fun  F  ->  ran 
F  e.  _V )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   _Vcvv 3078   dom cdm 4951   ran crn 4952   Fun wfun 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537
This theorem is referenced by:  zfrep6  6658  fornex  6659  tz7.48-3  7012  inf0  7942  noinfepOLD  7981  axcc2lem  8720  zorn2lem4  8783  hashimarn  12322  fnct  26191
  Copyright terms: Public domain W3C validator