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Theorem relrn0 5415
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
StepHypRef Expression
1 reldm0 5375 . 2 (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
2 dm0rn0 5374 . 2 (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
31, 2syl6bb 276 1 (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  c0 3948  dom cdm 5143  ran crn 5144  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  cnvsn0  5638  coeq0  5682  foconst  6164  fconst5  6512  edg0iedg0  25994  edg0iedg0OLD  25995  edg0usgr  26190  usgr1v0edg  26194  heicant  33574
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