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Theorem rn0 5614
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 5575 . 2 dom ∅ = ∅
2 dm0rn0 5578 . 2 (dom ∅ = ∅ ↔ ran ∅ = ∅)
31, 2mpbi 222 1 ran ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656  c0 4146  dom cdm 5346  ran crn 5347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-cnv 5354  df-dm 5356  df-rn 5357
This theorem is referenced by:  ima0  5726  0ima  5727  rnxpid  5812  xpima  5821  f0  6327  rnfvprc  6431  2ndval  7436  frxp  7556  oarec  7914  fodomr  8386  dfac5lem3  9268  itunitc  9565  0rest  16450  arwval  17052  psgnsn  18298  oppglsm  18415  mpfrcl  19885  ply1frcl  20050  edgval  26354  0grsubgr  26582  0uhgrsubgr  26583  0ngrp  27917  bafval  28010  locfinref  30449  esumrnmpt2  30671  sibf0  30937  mvtval  31939  mrsubvrs  31961  mstaval  31983  mzpmfp  38153  dmnonrel  38736  imanonrel  38739  conrel1d  38795  clsneibex  39239  neicvgbex  39249  sge00  41382
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