Theorem rn0 5245

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

Step	Hyp	Ref	Expression
1		dm0 5207	. 2 |- dom (/) = (/)
2		dm0rn0 5210	. 2 |- ( dom (/) = (/) <-> ran (/) = (/) )
3	1, 2	mpbi 208	1 |- ran (/) = (/)

Syntax hints:   = wceq 1374   (/)c0 3778   dom cdm 4992   ran crn 4993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-cnv 5000  df-dm 5002  df-rn 5003

This theorem is referenced by:  ima0  5343  0ima  5344  rnxpid  5431  xpima  5440  f0  5757  2ndval  6777  frxp  6883  oarec  7201  map0e  7446  fodomr  7658  dfac5lem3  8495  itunitc  8790  0rest  14674  arwval  15217  pmtrfrn  16272  psgnsn  16334  oppglsm  16451  mpfrcl  17951  ply1frcl  18119  nbgra0edg  24094  uvtx01vtx  24154  rusgra0edg  24617  0ngrp  24875  bafval  25159  sibf0  27902  mzpmfp  30270  mzpmfpOLD  30271  conrel1d  36656