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Theorem cnvdif 6096

Description: Distributive law for converse over class difference. (Contributed by Mario Carneiro, 26-Jun-2014.)

**Assertion**
Ref	Expression
cnvdif	⊢ ^◡(𝐴 ∖ 𝐵) = (^◡𝐴 ∖ ^◡𝐵)

Proof of Theorem cnvdif Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 6062	. 2 ⊢ Rel ^◡(𝐴 ∖ 𝐵)
2		difss 4068	. . 3 ⊢ (^◡𝐴 ∖ ^◡𝐵) ⊆ ^◡𝐴
3		relcnv 6062	. . 3 ⊢ Rel ^◡𝐴
4		relss 5727	. . 3 ⊢ ((^◡𝐴 ∖ ^◡𝐵) ⊆ ^◡𝐴 → (Rel ^◡𝐴 → Rel (^◡𝐴 ∖ ^◡𝐵)))
5	2, 3, 4	mp2 9	. 2 ⊢ Rel (^◡𝐴 ∖ ^◡𝐵)
6		eldif 3894	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7		vex 3437	. . . 4 ⊢ 𝑥 ∈ V
8		vex 3437	. . . 4 ⊢ 𝑦 ∈ V
9	7, 8	opelcnv 5825	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡(𝐴 ∖ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵))
10		eldif 3894	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (^◡𝐴 ∖ ^◡𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ^◡𝐵))
11	7, 8	opelcnv 5825	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
12	7, 8	opelcnv 5825	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
13	12	notbii 322	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ ^◡𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
14	11, 13	anbi12i 635	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ^◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ^◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
15	10, 14	bitri 277	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (^◡𝐴 ∖ ^◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
16	6, 9, 15	3bitr4i 305	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡(𝐴 ∖ 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (^◡𝐴 ∖ ^◡𝐵))
17	1, 5, 16	eqrelriiv 5735	1 ⊢ ^◡(𝐴 ∖ 𝐵) = (^◡𝐴 ∖ ^◡𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ∖ cdif 3881 ⊆ wss 3884 ⟨cop 4563 ^◡ccnv 5619 Rel wrel 5625

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-ext 2713 ax-sep 5220 ax-pr 5364

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-sb 2075 df-clab 2720 df-cleq 2733 df-clel 2816 df-rab 3394 df-v 3435 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-nul 4264 df-if 4457 df-sn 4558 df-pr 4560 df-op 4564 df-br 5075 df-opab 5137 df-xp 5626 df-rel 5627 df-cnv 5628

This theorem is referenced by: cnvin 6097 gtiso 32795 gsumhashmul 33150 mthmpps 35823 cnvnonrel 44045

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