Theorem cononrel2 44183

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

Assertion

Ref	Expression
cononrel2	⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅

Proof of Theorem cononrel2

Step	Hyp	Ref	Expression
1		cnvco 5866	. . . 4 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴)
2		cnvnonrel 44176	. . . . 5 ⊢ ◡(𝐵 ∖ ◡◡𝐵) = ∅
3	2	coeq1i 5836	. . . 4 ⊢ (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) = (∅ ∘ ◡𝐴)
4		co01 6253	. . . 4 ⊢ (∅ ∘ ◡𝐴) = ∅
5	1, 3, 4	3eqtri 2792	. . 3 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅
6	5	cnveqi 5851	. 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ◡∅
7		relco 6101	. . 3 ⊢ Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵))
8		dfrel2 6179	. . 3 ⊢ (Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) ↔ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)))
9	7, 8	mpbi 233	. 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵))
10		cnv0 5860	. 2 ⊢ ◡∅ = ∅
11	6, 9, 10	3eqtr3i 2796	1 ⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅

Syntax hints:   = wceq 1563   ∖ cdif 3904  ∅c0 4288  ^◡ccnv 5651   ∘ ccom 5656  Rel wrel 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-res 5664

This theorem is referenced by:  cnvtrcl0  44214