Theorem relrn0 4986

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 4941	. 2 |- ( Rel A -> ( A = (/) <-> dom A = (/) ) )
2		dm0rn0 4940	. 2 |- ( dom A = (/) <-> ran A = (/) )
3	1, 2	bitrdi 196	1 |- ( Rel A -> ( A = (/) <-> ran A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   (/)c0 3491   dom cdm 4719   ran crn 4720   Rel wrel 4724

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by:  cnvsn0  5197  edg0iedg0g  15866