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Theorem cononrel2 43608
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 5896 . . . 4 (𝐴 ∘ (𝐵𝐵)) = ((𝐵𝐵) ∘ 𝐴)
2 cnvnonrel 43601 . . . . 5 (𝐵𝐵) = ∅
32coeq1i 5870 . . . 4 ((𝐵𝐵) ∘ 𝐴) = (∅ ∘ 𝐴)
4 co01 6281 . . . 4 (∅ ∘ 𝐴) = ∅
51, 3, 43eqtri 2769 . . 3 (𝐴 ∘ (𝐵𝐵)) = ∅
65cnveqi 5885 . 2 (𝐴 ∘ (𝐵𝐵)) =
7 relco 6126 . . 3 Rel (𝐴 ∘ (𝐵𝐵))
8 dfrel2 6209 . . 3 (Rel (𝐴 ∘ (𝐵𝐵)) ↔ (𝐴 ∘ (𝐵𝐵)) = (𝐴 ∘ (𝐵𝐵)))
97, 8mpbi 230 . 2 (𝐴 ∘ (𝐵𝐵)) = (𝐴 ∘ (𝐵𝐵))
10 cnv0 6160 . 2 ∅ = ∅
116, 9, 103eqtr3i 2773 1 (𝐴 ∘ (𝐵𝐵)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cdif 3948  c0 4333  ccnv 5684  ccom 5689  Rel wrel 5690
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694
This theorem is referenced by:  cnvtrcl0  43639
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