Theorem cononrel2 44046

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 5834	. . . 4 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴)
2		cnvnonrel 44039	. . . . 5 ⊢ ◡(𝐵 ∖ ◡◡𝐵) = ∅
3	2	coeq1i 5808	. . . 4 ⊢ (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) = (∅ ∘ ◡𝐴)
4		co01 6220	. . . 4 ⊢ (∅ ∘ ◡𝐴) = ∅
5	1, 3, 4	3eqtri 2767	. . 3 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅
6	5	cnveqi 5823	. 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ◡∅
7		relco 6067	. . 3 ⊢ Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵))
8		dfrel2 6147	. . 3 ⊢ (Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) ↔ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)))
9	7, 8	mpbi 231	. 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵))
10		cnv0 6097	. 2 ⊢ ◡∅ = ∅
11	6, 9, 10	3eqtr3i 2771	1 ⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅

Syntax hints:   = wceq 1547   ∖ cdif 3887  ∅c0 4268  ^◡ccnv 5624   ∘ ccom 5629  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634

This theorem is referenced by:  cnvtrcl0  44077