Theorem rn0 5019

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 4976	. 2 |- dom (/) = (/)
2		dm0rn0 4979	. 2 |- ( dom (/) = (/) <-> ran (/) = (/) )
3	1, 2	mpbi 145	1 |- ran (/) = (/)

Syntax hints:   = wceq 1398   (/)c0 3512   dom cdm 4755   ran crn 4756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-br 4116  df-opab 4178  df-cnv 4763  df-dm 4765  df-rn 4766

This theorem is referenced by:  ima0  5127  0ima  5128  xpima1  5215  f0  5564  exmidfodomrlemim  7518  0grsubgr  16390  0uhgrsubgr  16391