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Theorem relrn0 5061
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 5020 . 2  |-  ( Rel 
A  ->  ( A  =  (/)  <->  dom  A  =  (/) ) )
2 dm0rn0 5019 . 2  |-  ( dom 
A  =  (/)  <->  ran  A  =  (/) )
31, 2syl6bb 253 1  |-  ( Rel 
A  ->  ( A  =  (/)  <->  ran  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   (/)c0 3564   dom cdm 4811   ran crn 4812   Rel wrel 4816
This theorem is referenced by:  cnvsn0  5271  foconst  5597  fconst5  5881  coeq0  26494
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-dm 4821  df-rn 4822
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