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Theorem relrn0 4796
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 4752 . 2 (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
2 dm0rn0 4751 . 2 (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
31, 2syl6bb 195 1 (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  c0 3358  dom cdm 4534  ran crn 4535  Rel wrel 4539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545
This theorem is referenced by:  cnvsn0  5002
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