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Theorem relrn0 5813
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 5771 . 2 (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
2 dm0rn0 5768 . 2 (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
31, 2syl6bb 290 1 (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  c0 4266  dom cdm 5528  ran crn 5529  Rel wrel 5533
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-xp 5534  df-rel 5535  df-cnv 5536  df-dm 5538  df-rn 5539
This theorem is referenced by:  cnvsn0  6040  coeq0  6081  foconst  6576  fconst5  6941  edg0iedg0  26826  edg0usgr  27021  usgr1v0edg  27025  tocyccntz  30793  heicant  34972
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