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Theorem rn0 4844
Description: The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
rn0  |-  ran  (/)  =  (/)

Proof of Theorem rn0
StepHypRef Expression
1 dm0 4802 . 2  |-  dom  (/)  =  (/)
2 dm0rn0 4805 . 2  |-  ( dom  (/)  =  (/)  <->  ran  (/)  =  (/) )
31, 2mpbi 144 1  |-  ran  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1335   (/)c0 3395   dom cdm 4588   ran crn 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-cnv 4596  df-dm 4598  df-rn 4599
This theorem is referenced by:  ima0  4947  0ima  4948  xpima1  5034  f0  5362  exmidfodomrlemim  7138
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