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Theorem rn0 4924
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 4880 . 2  |-  dom  (/)  =  (/)
2 dm0rn0 4883 . 2  |-  ( dom  (/)  =  (/)  <->  ran  (/)  =  (/) )
31, 2mpbi 201 1  |-  ran  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1619   (/)c0 3430   dom cdm 4661   ran crn 4662
This theorem is referenced by:  ima0  5018  0ima  5019  rnxpid  5097  f0  5363  2ndval  6059  frxp  6159  oarec  6528  map0e  6773  fodomr  6980  dfac5lem3  7720  itunitc  8015  0rest  13296  arwval  13837  oppglsm  14915  mpfrcl  19364  0ngrp  20838  bafval  21120  0alg  25123  mzpmfp  26192  pmtrfrn  26767
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-cnv 4677  df-dm 4679  df-rn 4680
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