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Theorem rn0 5789
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 5783 . 2 dom ∅ = ∅
2 dm0rn0 5788 . 2 (dom ∅ = ∅ ↔ ran ∅ = ∅)
31, 2mpbi 231 1 ran ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  c0 4288  dom cdm 5548  ran crn 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by:  ima0  5938  0ima  5939  rnxpid  6023  xpima  6032  f0  6553  rnfvprc  6657  2ndval  7681  frxp  7809  oarec  8177  fodomr  8656  dfac5lem3  9539  itunitc  9831  0rest  16691  arwval  17291  psgnsn  18577  oppglsm  18696  mpfrcl  20226  ply1frcl  20409  edgval  26761  0grsubgr  26987  0uhgrsubgr  26988  0ngrp  28215  bafval  28308  tocycf  30686  tocyc01  30687  locfinref  31004  esumrnmpt2  31226  sibf0  31491  mvtval  32644  mrsubvrs  32666  mstaval  32688  mzpmfp  39222  dmnonrel  39828  imanonrel  39831  conrel1d  39886  clsneibex  40330  neicvgbex  40340  sge00  42535
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