Theorem relrn0 5805

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 5762	. 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
2		dm0rn0 5759	. 2 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
3	1, 2	syl6bb 290	1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  ∅c0 4243  dom cdm 5519  ran crn 5520  Rel wrel 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530

This theorem is referenced by:  cnvsn0  6034  coeq0  6075  foconst  6578  fconst5  6945  edg0iedg0  26848  edg0usgr  27043  usgr1v0edg  27047  tocyccntz  30836  heicant  35092