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Theorem cononrel2 38395
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 5455 . . . 4 (𝐴 ∘ (𝐵𝐵)) = ((𝐵𝐵) ∘ 𝐴)
2 cnvnonrel 38388 . . . . 5 (𝐵𝐵) = ∅
32coeq1i 5429 . . . 4 ((𝐵𝐵) ∘ 𝐴) = (∅ ∘ 𝐴)
4 co01 5803 . . . 4 (∅ ∘ 𝐴) = ∅
51, 3, 43eqtri 2778 . . 3 (𝐴 ∘ (𝐵𝐵)) = ∅
65cnveqi 5444 . 2 (𝐴 ∘ (𝐵𝐵)) =
7 relco 5786 . . 3 Rel (𝐴 ∘ (𝐵𝐵))
8 dfrel2 5733 . . 3 (Rel (𝐴 ∘ (𝐵𝐵)) ↔ (𝐴 ∘ (𝐵𝐵)) = (𝐴 ∘ (𝐵𝐵)))
97, 8mpbi 220 . 2 (𝐴 ∘ (𝐵𝐵)) = (𝐴 ∘ (𝐵𝐵))
10 cnv0 5685 . 2 ∅ = ∅
116, 9, 103eqtr3i 2782 1 (𝐴 ∘ (𝐵𝐵)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1624  cdif 3704  c0 4050  ccnv 5257  ccom 5262  Rel wrel 5263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267
This theorem is referenced by:  cnvtrcl0  38427
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