Theorem relrn0 5920

Description: A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)

Assertion

Ref	Expression
relrn0	⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))

Proof of Theorem relrn0

Step	Hyp	Ref	Expression
1		reldm0 5875	. 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
2		dm0rn0 5871	. 2 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
3	1, 2	bitrdi 287	1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541  ∅c0 4283  dom cdm 5622  ran crn 5623  Rel wrel 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633

This theorem is referenced by:  cnvsn0  6166  coeq0  6212  foconst  6759  fconst5  7150  cnvfi  9098  edg0iedg0  29077  edg0usgr  29275  usgr1v0edg  29279  tocyccntz  33175  1arithidom  33567  heicant  37795  tfsconcat00  43531