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Theorem cononrel2 38427
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 5446 . . . 4 (𝐴 ∘ (𝐵𝐵)) = ((𝐵𝐵) ∘ 𝐴)
2 cnvnonrel 38420 . . . . 5 (𝐵𝐵) = ∅
32coeq1i 5420 . . . 4 ((𝐵𝐵) ∘ 𝐴) = (∅ ∘ 𝐴)
4 co01 5794 . . . 4 (∅ ∘ 𝐴) = ∅
51, 3, 43eqtri 2797 . . 3 (𝐴 ∘ (𝐵𝐵)) = ∅
65cnveqi 5435 . 2 (𝐴 ∘ (𝐵𝐵)) =
7 relco 5777 . . 3 Rel (𝐴 ∘ (𝐵𝐵))
8 dfrel2 5724 . . 3 (Rel (𝐴 ∘ (𝐵𝐵)) ↔ (𝐴 ∘ (𝐵𝐵)) = (𝐴 ∘ (𝐵𝐵)))
97, 8mpbi 220 . 2 (𝐴 ∘ (𝐵𝐵)) = (𝐴 ∘ (𝐵𝐵))
10 cnv0 5676 . 2 ∅ = ∅
116, 9, 103eqtr3i 2801 1 (𝐴 ∘ (𝐵𝐵)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cdif 3720  c0 4063  ccnv 5248  ccom 5253  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258
This theorem is referenced by:  cnvtrcl0  38459
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