Theorem relrn0 4612

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

Step	Hyp	Ref	Expression
1		reldm0 4571	. 2 |- ( Rel A -> ( A = (/) <-> dom A = (/) ) )
2		dm0rn0 4570	. 2 |- ( dom A = (/) <-> ran A = (/) )
3	1, 2	syl6bb 194	1 |- ( Rel A -> ( A = (/) <-> ran A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   (/)c0 3251   dom cdm 4363   ran crn 4364   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by:  cnvsn0  4809