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Theorem rn0 5127
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 5083 . 2  |-  dom  (/)  =  (/)
2 dm0rn0 5086 . 2  |-  ( dom  (/)  =  (/)  <->  ran  (/)  =  (/) )
31, 2mpbi 200 1  |-  ran  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1652   (/)c0 3628   dom cdm 4878   ran crn 4879
This theorem is referenced by:  ima0  5221  0ima  5222  rnxpid  5302  xpima  5313  f0  5627  2ndval  6352  frxp  6456  oarec  6805  map0e  7051  fodomr  7258  dfac5lem3  8006  itunitc  8301  0rest  13657  arwval  14198  oppglsm  15276  mpfrcl  19939  nbgra0edg  21444  uvtx01vtx  21501  0ngrp  21799  bafval  22083  sibf0  24649  mzpmfp  26804  pmtrfrn  27377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-cnv 4886  df-dm 4888  df-rn 4889
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