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Theorem rn0 5926
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 5921 . 2 dom ∅ = ∅
2 dm0rn0 5925 . 2 (dom ∅ = ∅ ↔ ran ∅ = ∅)
31, 2mpbi 229 1 ran ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  c0 4323  dom cdm 5677  ran crn 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-cnv 5685  df-dm 5687  df-rn 5688
This theorem is referenced by:  ima0  6077  0ima  6078  rnxpid  6173  xpima  6182  f0  6773  rnfvprc  6886  2ndval  7978  frxp  8112  oarec  8562  fodomr  9128  dfac5lem3  10120  itunitc  10416  relexprnd  14995  0rest  17375  arwval  17993  psgnsn  19388  oppglsm  19510  mpfrcl  21648  ply1frcl  21837  edgval  28309  0grsubgr  28535  0uhgrsubgr  28536  0ngrp  29764  bafval  29857  tocycf  32276  tocyc01  32277  locfinref  32821  esumrnmpt2  33066  sibf0  33333  mvtval  34491  mrsubvrs  34513  mstaval  34535  mzpmfp  41485  dmnonrel  42341  imanonrel  42344  conrel1d  42414  clsneibex  42853  neicvgbex  42863  sge00  45092
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