Theorem cononrel1 43024

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 5888	. . . 4 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴))
2		cnvnonrel 43018	. . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
3	2	coeq2i 5863	. . . 4 ⊢ (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴)) = (◡𝐵 ∘ ∅)
4		co02 6264	. . . 4 ⊢ (◡𝐵 ∘ ∅) = ∅
5	1, 3, 4	3eqtri 2760	. . 3 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅
6	5	cnveqi 5877	. 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ◡∅
7		relco 6112	. . 3 ⊢ Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵)
8		dfrel2 6193	. . 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 6145	. 2 ⊢ ◡∅ = ∅
11	6, 9, 10	3eqtr3i 2764	1 ⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅

Syntax hints:   = wceq 1534   ∖ cdif 3944  ∅c0 4323  ^◡ccnv 5677   ∘ ccom 5682  Rel wrel 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687

This theorem is referenced by:  cnvtrcl0  43056