Theorem rn0 4721

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 4681	. 2 |- dom (/) = (/)
2		dm0rn0 4684	. 2 |- ( dom (/) = (/) <-> ran (/) = (/) )
3	1, 2	mpbi 144	1 |- ran (/) = (/)

Syntax hints:   = wceq 1296   (/)c0 3302   dom cdm 4467   ran crn 4468

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-cnv 4475  df-dm 4477  df-rn 4478

This theorem is referenced by:  ima0  4824  0ima  4825  xpima1  4911  f0  5236  exmidfodomrlemim  6924