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Theorem cononrel2 43090
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 5882 . . . 4 (𝐴 ∘ (𝐵𝐵)) = ((𝐵𝐵) ∘ 𝐴)
2 cnvnonrel 43083 . . . . 5 (𝐵𝐵) = ∅
32coeq1i 5856 . . . 4 ((𝐵𝐵) ∘ 𝐴) = (∅ ∘ 𝐴)
4 co01 6260 . . . 4 (∅ ∘ 𝐴) = ∅
51, 3, 43eqtri 2757 . . 3 (𝐴 ∘ (𝐵𝐵)) = ∅
65cnveqi 5871 . 2 (𝐴 ∘ (𝐵𝐵)) =
7 relco 6107 . . 3 Rel (𝐴 ∘ (𝐵𝐵))
8 dfrel2 6188 . . 3 (Rel (𝐴 ∘ (𝐵𝐵)) ↔ (𝐴 ∘ (𝐵𝐵)) = (𝐴 ∘ (𝐵𝐵)))
97, 8mpbi 229 . 2 (𝐴 ∘ (𝐵𝐵)) = (𝐴 ∘ (𝐵𝐵))
10 cnv0 6140 . 2 ∅ = ∅
116, 9, 103eqtr3i 2761 1 (𝐴 ∘ (𝐵𝐵)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cdif 3936  c0 4318  ccnv 5671  ccom 5676  Rel wrel 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681
This theorem is referenced by:  cnvtrcl0  43121
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