Theorem rn0 4901

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 4859	. 2 ⊢ dom ∅ = ∅
2		dm0rn0 4862	. 2 ⊢ (dom ∅ = ∅ ↔ ran ∅ = ∅)
3	1, 2	mpbi 145	1 ⊢ ran ∅ = ∅

Syntax hints:   = wceq 1364  ∅c0 3437  dom cdm 4644  ran crn 4645

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-cnv 4652  df-dm 4654  df-rn 4655

This theorem is referenced by:  ima0  5005  0ima  5006  xpima1  5093  f0  5425  exmidfodomrlemim  7230