MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relrn0 Unicode version

Theorem relrn0 4937
Description: A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
relrn0  |-  ( Rel 
A  ->  ( A  =  (/)  <->  ran  A  =  (/) ) )

Proof of Theorem relrn0
StepHypRef Expression
1 reldm0 4896 . 2  |-  ( Rel 
A  ->  ( A  =  (/)  <->  dom  A  =  (/) ) )
2 dm0rn0 4895 . 2  |-  ( dom 
A  =  (/)  <->  ran  A  =  (/) )
31, 2syl6bb 252 1  |-  ( Rel 
A  ->  ( A  =  (/)  <->  ran  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   (/)c0 3455   dom cdm 4689   ran crn 4690   Rel wrel 4694
This theorem is referenced by:  cnvsn0  5141  foconst  5462  fconst5  5731  coeq0  26831
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 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
  Copyright terms: Public domain W3C validator