Theorem rn0 4923

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 4881	. 2 |- dom (/) = (/)
2		dm0rn0 4884	. 2 |- ( dom (/) = (/) <-> ran (/) = (/) )
3	1, 2	mpbi 145	1 |- ran (/) = (/)

Syntax hints:   = wceq 1364   (/)c0 3451   dom cdm 4664   ran crn 4665

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-cnv 4672  df-dm 4674  df-rn 4675

This theorem is referenced by:  ima0  5029  0ima  5030  xpima1  5117  f0  5451  exmidfodomrlemim  7280