Theorem rn0 4932

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 4890	. 2 |- dom (/) = (/)
2		dm0rn0 4893	. 2 |- ( dom (/) = (/) <-> ran (/) = (/) )
3	1, 2	mpbi 145	1 |- ran (/) = (/)

Syntax hints:   = wceq 1372   (/)c0 3459   dom cdm 4673   ran crn 4674

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-cnv 4681  df-dm 4683  df-rn 4684

This theorem is referenced by:  ima0  5038  0ima  5039  xpima1  5126  f0  5460  exmidfodomrlemim  7291