Theorem relrn0 4594

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 4553	. 2 |- ( Rel A -> ( A = (/) <-> dom A = (/) ) )
2		dm0rn0 4552	. 2 |- ( dom A = (/) <-> ran A = (/) )
3	1, 2	syl6bb 185	1 |- ( Rel A -> ( A = (/) <-> ran A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   (/)c0 3224   dom cdm 4345   ran crn 4346   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  cnvsn0  4789