Theorem cononrel2 44361

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 5880	. . . 4 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴)
2		cnvnonrel 44354	. . . . 5 ⊢ ◡(𝐵 ∖ ◡◡𝐵) = ∅
3	2	coeq1i 5850	. . . 4 ⊢ (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) = (∅ ∘ ◡𝐴)
4		co01 6268	. . . 4 ⊢ (∅ ∘ ◡𝐴) = ∅
5	1, 3, 4	3eqtri 2793	. . 3 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅
6	5	cnveqi 5865	. 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ◡∅
7		relco 6115	. . 3 ⊢ Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵))
8		dfrel2 6192	. . 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 5874	. 2 ⊢ ◡∅ = ∅
11	6, 9, 10	3eqtr3i 2797	1 ⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 3905  ∅c0 4289  ^◡ccnv 5665   ∘ ccom 5670  Rel wrel 5671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-res 5678

This theorem is used by:  cnvtrcl0  44392