Theorem rn0 3355

Description: The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36.

Assertion

Ref	Expression
rn0	|- ran (/) = (/)

Proof of Theorem rn0

Step	Hyp	Ref	Expression
1		dm0 3323	. 2 |- dom (/) = (/)
2		dm0rn0 3330	. 2 |- (dom (/) = (/) <-> ran (/) = (/))
3	1, 2	mpbi 189	1 |- ran (/) = (/)

Syntax hints:   = wceq 956  (/)c0 2280  dom cdm 3170  ran crn 3171

This theorem is referenced by:  ima0 3420  0ima 3421  rnxpss 3474  f0 3656  2ndval 4082  map0e 4342  fodomr 4483  aceq5lem3 4737  infxpidmlem4 7555  infxpidmlem8 7559  infxpidmlem10 7561  0ngrp 8055  bafval 8223  0alg 10689

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-cnv 3186  df-dm 3188  df-rn 3189

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