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Theorem rn0 5244
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 5206 . 2  |-  dom  (/)  =  (/)
2 dm0rn0 5209 . 2  |-  ( dom  (/)  =  (/)  <->  ran  (/)  =  (/) )
31, 2mpbi 208 1  |-  ran  (/)  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383   (/)c0 3770   dom cdm 4989   ran crn 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-cnv 4997  df-dm 4999  df-rn 5000
This theorem is referenced by:  ima0  5342  0ima  5343  rnxpid  5430  xpima  5439  f0  5756  2ndval  6788  frxp  6895  oarec  7213  fodomr  7670  dfac5lem3  8509  itunitc  8804  0rest  14808  arwval  15348  pmtrfrn  16461  psgnsn  16523  oppglsm  16640  mpfrcl  18165  ply1frcl  18333  nbgra0edg  24408  uvtx01vtx  24468  rusgra0edg  24931  0ngrp  25189  bafval  25473  locfinref  27821  sibf0  28253  mvtval  28837  mrsubrn  28850  mrsub0  28853  mrsubf  28854  mrsubccat  28855  mrsubcn  28856  mrsubco  28858  mrsubvrs  28859  elmsubrn  28865  msubrn  28866  msubf  28869  mstaval  28881  mzpmfp  30654  mzpmfpOLD  30655  conrel1d  37464
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