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Theorem cononrel2 43585
Description: Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
Assertion
Ref Expression
cononrel2 (𝐴 ∘ (𝐵𝐵)) = ∅

Proof of Theorem cononrel2
StepHypRef Expression
1 cnvco 5899 . . . 4 (𝐴 ∘ (𝐵𝐵)) = ((𝐵𝐵) ∘ 𝐴)
2 cnvnonrel 43578 . . . . 5 (𝐵𝐵) = ∅
32coeq1i 5873 . . . 4 ((𝐵𝐵) ∘ 𝐴) = (∅ ∘ 𝐴)
4 co01 6283 . . . 4 (∅ ∘ 𝐴) = ∅
51, 3, 43eqtri 2767 . . 3 (𝐴 ∘ (𝐵𝐵)) = ∅
65cnveqi 5888 . 2 (𝐴 ∘ (𝐵𝐵)) =
7 relco 6129 . . 3 Rel (𝐴 ∘ (𝐵𝐵))
8 dfrel2 6211 . . 3 (Rel (𝐴 ∘ (𝐵𝐵)) ↔ (𝐴 ∘ (𝐵𝐵)) = (𝐴 ∘ (𝐵𝐵)))
97, 8mpbi 230 . 2 (𝐴 ∘ (𝐵𝐵)) = (𝐴 ∘ (𝐵𝐵))
10 cnv0 6163 . 2 ∅ = ∅
116, 9, 103eqtr3i 2771 1 (𝐴 ∘ (𝐵𝐵)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cdif 3960  c0 4339  ccnv 5688  ccom 5693  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698
This theorem is referenced by:  cnvtrcl0  43616
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