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Theorem cononrel1 40287
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 5724 . . . 4 ((𝐴𝐴) ∘ 𝐵) = (𝐵(𝐴𝐴))
2 cnvnonrel 40281 . . . . 5 (𝐴𝐴) = ∅
32coeq2i 5699 . . . 4 (𝐵(𝐴𝐴)) = (𝐵 ∘ ∅)
4 co02 6084 . . . 4 (𝐵 ∘ ∅) = ∅
51, 3, 43eqtri 2828 . . 3 ((𝐴𝐴) ∘ 𝐵) = ∅
65cnveqi 5713 . 2 ((𝐴𝐴) ∘ 𝐵) =
7 relco 6068 . . 3 Rel ((𝐴𝐴) ∘ 𝐵)
8 dfrel2 6017 . . 3 (Rel ((𝐴𝐴) ∘ 𝐵) ↔ ((𝐴𝐴) ∘ 𝐵) = ((𝐴𝐴) ∘ 𝐵))
97, 8mpbi 233 . 2 ((𝐴𝐴) ∘ 𝐵) = ((𝐴𝐴) ∘ 𝐵)
10 cnv0 5970 . 2 ∅ = ∅
116, 9, 103eqtr3i 2832 1 ((𝐴𝐴) ∘ 𝐵) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cdif 3881  c0 4246  ccnv 5522  ccom 5527  Rel wrel 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532
This theorem is referenced by:  cnvtrcl0  40319
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