Theorem relrn0 5958

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 5917	. 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
2		dm0rn0 5914	. 2 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
3	1, 2	bitrdi 287	1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533  ∅c0 4314  dom cdm 5666  ran crn 5667  Rel wrel 5671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677

This theorem is referenced by:  cnvsn0  6199  coeq0  6244  foconst  6810  fconst5  7199  cnvfi  9175  edg0iedg0  28750  edg0usgr  28945  usgr1v0edg  28949  tocyccntz  32737  heicant  36979  tfsconcat00  42552