Theorem cononrel1 43089

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 5882	. . . 4 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴))
2		cnvnonrel 43083	. . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
3	2	coeq2i 5857	. . . 4 ⊢ (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴)) = (◡𝐵 ∘ ∅)
4		co02 6259	. . . 4 ⊢ (◡𝐵 ∘ ∅) = ∅
5	1, 3, 4	3eqtri 2757	. . 3 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅
6	5	cnveqi 5871	. 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ◡∅
7		relco 6107	. . 3 ⊢ Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵)
8		dfrel2 6188	. . 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 6140	. 2 ⊢ ◡∅ = ∅
11	6, 9, 10	3eqtr3i 2761	1 ⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅

Syntax hints:   = wceq 1533   ∖ cdif 3936  ∅c0 4318  ^◡ccnv 5671   ∘ ccom 5676  Rel wrel 5677

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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681

This theorem is referenced by:  cnvtrcl0  43121