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Theorem dmnonrel 41236
Description: The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
Assertion
Ref Expression
dmnonrel dom (𝐴𝐴) = ∅

Proof of Theorem dmnonrel
StepHypRef Expression
1 dfdm4 5817 . 2 dom (𝐴𝐴) = ran (𝐴𝐴)
2 cnvnonrel 41234 . . 3 (𝐴𝐴) = ∅
32rneqi 5858 . 2 ran (𝐴𝐴) = ran ∅
4 rn0 5847 . 2 ran ∅ = ∅
51, 3, 43eqtri 2768 1 dom (𝐴𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cdif 3889  c0 4262  ccnv 5599  dom cdm 5600  ran crn 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-xp 5606  df-rel 5607  df-cnv 5608  df-dm 5610  df-rn 5611
This theorem is referenced by:  rnnonrel  41237
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