Theorem cononrel1 42921

Description: Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)

Assertion

Ref	Expression
cononrel1	⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅

Proof of Theorem cononrel1

Step	Hyp	Ref	Expression
1		cnvco 5879	. . . 4 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴))
2		cnvnonrel 42915	. . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
3	2	coeq2i 5854	. . . 4 ⊢ (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴)) = (◡𝐵 ∘ ∅)
4		co02 6253	. . . 4 ⊢ (◡𝐵 ∘ ∅) = ∅
5	1, 3, 4	3eqtri 2758	. . 3 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅
6	5	cnveqi 5868	. 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ◡∅
7		relco 6101	. . 3 ⊢ Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵)
8		dfrel2 6182	. . 3 ⊢ (Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) ↔ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵))
9	7, 8	mpbi 229	. 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵)
10		cnv0 6134	. 2 ⊢ ◡∅ = ∅
11	6, 9, 10	3eqtr3i 2762	1 ⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅

Syntax hints:   = wceq 1533   ∖ cdif 3940  ∅c0 4317  ^◡ccnv 5668   ∘ ccom 5673  Rel wrel 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678

This theorem is referenced by:  cnvtrcl0  42953