Theorem cononrel1 43712

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 5829	. . . 4 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴))
2		cnvnonrel 43706	. . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
3	2	coeq2i 5804	. . . 4 ⊢ (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴)) = (◡𝐵 ∘ ∅)
4		co02 6213	. . . 4 ⊢ (◡𝐵 ∘ ∅) = ∅
5	1, 3, 4	3eqtri 2760	. . 3 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅
6	5	cnveqi 5818	. 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ◡∅
7		relco 6061	. . 3 ⊢ Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵)
8		dfrel2 6141	. . 3 ⊢ (Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) ↔ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵))
9	7, 8	mpbi 230	. 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵)
10		cnv0 6091	. 2 ⊢ ◡∅ = ∅
11	6, 9, 10	3eqtr3i 2764	1 ⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅

Syntax hints:   = wceq 1541   ∖ cdif 3895  ∅c0 4282  ^◡ccnv 5618   ∘ ccom 5623  Rel wrel 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628

This theorem is referenced by:  cnvtrcl0  43744