Theorem relrn0 3350

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 3326	. 2 |- (Rel A -> (A = (/) <-> dom A = (/)))
2		dm0rn0 3325	. 2 |- (dom A = (/) <-> ran A = (/))
3	1, 2	syl6bb 535	1 |- (Rel A -> (A = (/) <-> ran A = (/)))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 954  (/)c0 2276  dom cdm 3165  ran crn 3166  Rel wrel 3170

This theorem is referenced by:  foconst 3674  fconst5 3839  infxpidmlem11 7513

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-cnv 3181  df-dm 3183  df-rn 3184

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