Theorem rn0 4956

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 4914	. 2 ⊢ dom ∅ = ∅
2		dm0rn0 4917	. 2 ⊢ (dom ∅ = ∅ ↔ ran ∅ = ∅)
3	1, 2	mpbi 145	1 ⊢ ran ∅ = ∅

Syntax hints:   = wceq 1375  ∅c0 3471  dom cdm 4696  ran crn 4697

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-cnv 4704  df-dm 4706  df-rn 4707

This theorem is referenced by:  ima0  5063  0ima  5064  xpima1  5151  f0  5492  exmidfodomrlemim  7347