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Theorem rn0 5935
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 5930 . 2 dom ∅ = ∅
2 dm0rn0 5934 . 2 (dom ∅ = ∅ ↔ ran ∅ = ∅)
31, 2mpbi 230 1 ran ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  c0 4332  dom cdm 5684  ran crn 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-cnv 5692  df-dm 5694  df-rn 5695
This theorem is referenced by:  ima0  6094  0ima  6095  rnxpid  6192  xpima  6201  f0  6788  rnfvprc  6899  2ndval  8018  frxp  8152  oarec  8601  fodomr  9169  fodomfir  9369  dfac5lem3  10166  itunitc  10462  relexprnd  15088  0rest  17475  arwval  18089  psgnsn  19539  oppglsm  19661  mpfrcl  22110  ply1frcl  22323  edgval  29067  0grsubgr  29296  0uhgrsubgr  29297  0ngrp  30531  bafval  30624  tocycf  33138  tocyc01  33139  unitprodclb  33418  locfinref  33841  esumrnmpt2  34070  sibf0  34337  mvtval  35506  mrsubvrs  35528  mstaval  35550  mzpmfp  42763  dmnonrel  43608  imanonrel  43611  conrel1d  43681  clsneibex  44120  neicvgbex  44130  sge00  46396
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