Theorem cononrel1 36919

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 5230	. . . 4 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴))
2		cnvnonrel 36913	. . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
3	2	coeq2i 5204	. . . 4 ⊢ (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴)) = (◡𝐵 ∘ ∅)
4		co02 5566	. . . 4 ⊢ (◡𝐵 ∘ ∅) = ∅
5	1, 3, 4	3eqtri 2636	. . 3 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅
6	5	cnveqi 5219	. 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ◡∅
7		relco 5550	. . 3 ⊢ Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵)
8		dfrel2 5502	. . 3 ⊢ (Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) ↔ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵))
9	7, 8	mpbi 219	. 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵)
10		cnv0 5454	. 2 ⊢ ◡∅ = ∅
11	6, 9, 10	3eqtr3i 2640	1 ⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅

Syntax hints:   = wceq 1475   ∖ cdif 3537  ∅c0 3874  ^◡ccnv 5037   ∘ ccom 5042  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047

This theorem is referenced by:  cnvtrcl0  36952