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Theorem cononrel1 38421
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 5464 . . . 4 ((𝐴𝐴) ∘ 𝐵) = (𝐵(𝐴𝐴))
2 cnvnonrel 38415 . . . . 5 (𝐴𝐴) = ∅
32coeq2i 5439 . . . 4 (𝐵(𝐴𝐴)) = (𝐵 ∘ ∅)
4 co02 5811 . . . 4 (𝐵 ∘ ∅) = ∅
51, 3, 43eqtri 2787 . . 3 ((𝐴𝐴) ∘ 𝐵) = ∅
65cnveqi 5453 . 2 ((𝐴𝐴) ∘ 𝐵) =
7 relco 5795 . . 3 Rel ((𝐴𝐴) ∘ 𝐵)
8 dfrel2 5742 . . 3 (Rel ((𝐴𝐴) ∘ 𝐵) ↔ ((𝐴𝐴) ∘ 𝐵) = ((𝐴𝐴) ∘ 𝐵))
97, 8mpbi 220 . 2 ((𝐴𝐴) ∘ 𝐵) = ((𝐴𝐴) ∘ 𝐵)
10 cnv0 5694 . 2 ∅ = ∅
116, 9, 103eqtr3i 2791 1 ((𝐴𝐴) ∘ 𝐵) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  cdif 3713  c0 4059  ccnv 5266  ccom 5271  Rel wrel 5272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276
This theorem is referenced by:  cnvtrcl0  38454
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