Theorem relrn0 4907

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 4863	. 2 |- ( Rel A -> ( A = (/) <-> dom A = (/) ) )
2		dm0rn0 4862	. 2 |- ( dom A = (/) <-> ran A = (/) )
3	1, 2	bitrdi 196	1 |- ( Rel A -> ( A = (/) <-> ran A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   (/)c0 3437   dom cdm 4644   ran crn 4645   Rel wrel 4649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655

This theorem is referenced by:  cnvsn0  5115