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Theorem cononrel1 38670
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 5509 . . . 4 ((𝐴𝐴) ∘ 𝐵) = (𝐵(𝐴𝐴))
2 cnvnonrel 38664 . . . . 5 (𝐴𝐴) = ∅
32coeq2i 5484 . . . 4 (𝐵(𝐴𝐴)) = (𝐵 ∘ ∅)
4 co02 5866 . . . 4 (𝐵 ∘ ∅) = ∅
51, 3, 43eqtri 2823 . . 3 ((𝐴𝐴) ∘ 𝐵) = ∅
65cnveqi 5498 . 2 ((𝐴𝐴) ∘ 𝐵) =
7 relco 5850 . . 3 Rel ((𝐴𝐴) ∘ 𝐵)
8 dfrel2 5798 . . 3 (Rel ((𝐴𝐴) ∘ 𝐵) ↔ ((𝐴𝐴) ∘ 𝐵) = ((𝐴𝐴) ∘ 𝐵))
97, 8mpbi 222 . 2 ((𝐴𝐴) ∘ 𝐵) = ((𝐴𝐴) ∘ 𝐵)
10 cnv0 5751 . 2 ∅ = ∅
116, 9, 103eqtr3i 2827 1 ((𝐴𝐴) ∘ 𝐵) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  cdif 3764  c0 4113  ccnv 5309  ccom 5314  Rel wrel 5315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319
This theorem is referenced by:  cnvtrcl0  38703
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