Theorem relrn0 2568

Description: A relation is empty iff its range is empty.

Assertion

Ref	Expression
relrn0	|- (Rel A -> (A = (/) <-> ran A = (/)))

Proof of Theorem relrn0

Step	Hyp	Ref	Expression
1		reldm0 2550	. 2 |- (Rel A -> (A = (/) <-> dom A = (/)))
2		dm0rn0 2549	. 2 |- (dom A = (/) <-> ran A = (/))
3	1, 2	syl6bb 414	1 |- (Rel A -> (A = (/) <-> ran A = (/)))

Syntax hints:   -> wi 2   <-> wb 127   = wceq 1091  (/)c0 1707  dom cdm 2410  ran crn 2411  Rel wrel 2415

This theorem is referenced by:  infxpidmlem11 4943

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429

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