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Theorem rn0 5932
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 5927 . 2 dom ∅ = ∅
2 dm0rn0 5931 . 2 (dom ∅ = ∅ ↔ ran ∅ = ∅)
31, 2mpbi 229 1 ran ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  c0 4325  dom cdm 5682  ran crn 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-cnv 5690  df-dm 5692  df-rn 5693
This theorem is referenced by:  ima0  6086  0ima  6087  rnxpid  6184  xpima  6193  f0  6783  rnfvprc  6895  2ndval  8006  frxp  8140  oarec  8592  fodomr  9166  fodomfir  9370  dfac5lem3  10168  itunitc  10464  relexprnd  15053  0rest  17444  arwval  18065  psgnsn  19518  oppglsm  19640  mpfrcl  22100  ply1frcl  22309  edgval  28985  0grsubgr  29214  0uhgrsubgr  29215  0ngrp  30444  bafval  30537  tocycf  32995  tocyc01  32996  unitprodclb  33264  locfinref  33656  esumrnmpt2  33901  sibf0  34168  mvtval  35328  mrsubvrs  35350  mstaval  35372  mzpmfp  42404  dmnonrel  43257  imanonrel  43260  conrel1d  43330  clsneibex  43769  neicvgbex  43779  sge00  45997
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