Theorem cononrel2 44301

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 5877	. . . 4 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴)
2		cnvnonrel 44294	. . . . 5 ⊢ ◡(𝐵 ∖ ◡◡𝐵) = ∅
3	2	coeq1i 5847	. . . 4 ⊢ (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) = (∅ ∘ ◡𝐴)
4		co01 6265	. . . 4 ⊢ (∅ ∘ ◡𝐴) = ∅
5	1, 3, 4	3eqtri 2790	. . 3 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅
6	5	cnveqi 5862	. 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ◡∅
7		relco 6112	. . 3 ⊢ Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵))
8		dfrel2 6189	. . 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 5871	. 2 ⊢ ◡∅ = ∅
11	6, 9, 10	3eqtr3i 2794	1 ⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅

Syntax hints:   = wceq 1570   ∖ cdif 3903  ∅c0 4287  ^◡ccnv 5662   ∘ ccom 5667  Rel wrel 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-res 5675

This theorem is referenced by:  cnvtrcl0  44332