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Theorem rn0 4935
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 4891 . 2  |-  dom  (/)  =  (/)
2 dm0rn0 4894 . 2  |-  ( dom  (/)  =  (/)  <->  ran  (/)  =  (/) )
31, 2mpbi 201 1  |-  ran  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1624   (/)c0 3456   dom cdm 4688   ran crn 4689
This theorem is referenced by:  ima0  5029  0ima  5030  rnxpid  5108  f0  5390  2ndval  6086  frxp  6186  oarec  6555  map0e  6800  fodomr  7007  dfac5lem3  7747  itunitc  8042  0rest  13328  arwval  13869  oppglsm  14947  mpfrcl  19396  0ngrp  20870  bafval  21152  0alg  25155  mzpmfp  26224  pmtrfrn  26799
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-cnv 4696  df-dm 4698  df-rn 4699
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