Theorem rn0 5192

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 5154	. 2 |- dom (/) = (/)
2		dm0rn0 5157	. 2 |- ( dom (/) = (/) <-> ran (/) = (/) )
3	1, 2	mpbi 208	1 |- ran (/) = (/)

Syntax hints:   = wceq 1370   (/)c0 3738   dom cdm 4941   ran crn 4942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-cnv 4949  df-dm 4951  df-rn 4952

This theorem is referenced by:  ima0  5285  0ima  5286  rnxpid  5372  xpima  5381  f0  5693  2ndval  6683  frxp  6785  oarec  7104  map0e  7353  fodomr  7565  dfac5lem3  8399  itunitc  8694  0rest  14479  arwval  15022  pmtrfrn  16075  psgnsn  16137  oppglsm  16254  mpfrcl  17720  ply1frcl  17871  nbgra0edg  23488  uvtx01vtx  23545  0ngrp  23843  bafval  24127  sibf0  26857  mzpmfp  29224  mzpmfpOLD  29225  rusgra0edg  30714