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Theorem cononrel1 43024
Description: Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
Assertion
Ref Expression
cononrel1 ((𝐴𝐴) ∘ 𝐵) = ∅

Proof of Theorem cononrel1
StepHypRef Expression
1 cnvco 5888 . . . 4 ((𝐴𝐴) ∘ 𝐵) = (𝐵(𝐴𝐴))
2 cnvnonrel 43018 . . . . 5 (𝐴𝐴) = ∅
32coeq2i 5863 . . . 4 (𝐵(𝐴𝐴)) = (𝐵 ∘ ∅)
4 co02 6264 . . . 4 (𝐵 ∘ ∅) = ∅
51, 3, 43eqtri 2760 . . 3 ((𝐴𝐴) ∘ 𝐵) = ∅
65cnveqi 5877 . 2 ((𝐴𝐴) ∘ 𝐵) =
7 relco 6112 . . 3 Rel ((𝐴𝐴) ∘ 𝐵)
8 dfrel2 6193 . . 3 (Rel ((𝐴𝐴) ∘ 𝐵) ↔ ((𝐴𝐴) ∘ 𝐵) = ((𝐴𝐴) ∘ 𝐵))
97, 8mpbi 229 . 2 ((𝐴𝐴) ∘ 𝐵) = ((𝐴𝐴) ∘ 𝐵)
10 cnv0 6145 . 2 ∅ = ∅
116, 9, 103eqtr3i 2764 1 ((𝐴𝐴) ∘ 𝐵) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  cdif 3944  c0 4323  ccnv 5677  ccom 5682  Rel wrel 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687
This theorem is referenced by:  cnvtrcl0  43056
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