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Theorem cnvdif 6116

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 6075	. 2 ⊢ Rel ^◡(𝐴 ∖ 𝐵)
2		difss 4099	. . 3 ⊢ (^◡𝐴 ∖ ^◡𝐵) ⊆ ^◡𝐴
3		relcnv 6075	. . 3 ⊢ Rel ^◡𝐴
4		relss 5744	. . 3 ⊢ ((^◡𝐴 ∖ ^◡𝐵) ⊆ ^◡𝐴 → (Rel ^◡𝐴 → Rel (^◡𝐴 ∖ ^◡𝐵)))
5	2, 3, 4	mp2 9	. 2 ⊢ Rel (^◡𝐴 ∖ ^◡𝐵)
6		eldif 3924	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7		vex 3451	. . . 4 ⊢ 𝑥 ∈ V
8		vex 3451	. . . 4 ⊢ 𝑦 ∈ V
9	7, 8	opelcnv 5845	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡(𝐴 ∖ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵))
10		eldif 3924	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (^◡𝐴 ∖ ^◡𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ^◡𝐵))
11	7, 8	opelcnv 5845	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
12	7, 8	opelcnv 5845	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
13	12	notbii 320	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ ^◡𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
14	11, 13	anbi12i 628	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ^◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ^◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
15	10, 14	bitri 275	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (^◡𝐴 ∖ ^◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
16	6, 9, 15	3bitr4i 303	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡(𝐴 ∖ 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (^◡𝐴 ∖ ^◡𝐵))
17	1, 5, 16	eqrelriiv 5753	1 ⊢ ^◡(𝐴 ∖ 𝐵) = (^◡𝐴 ∖ ^◡𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∖ cdif 3911 ⊆ wss 3914 ⟨cop 4595 ^◡ccnv 5637 Rel wrel 5643

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-ss 3931 df-nul 4297 df-if 4489 df-sn 4590 df-pr 4592 df-op 4596 df-br 5108 df-opab 5170 df-xp 5644 df-rel 5645 df-cnv 5646

This theorem is referenced by: cnvin 6117 gtiso 32624 gsumhashmul 33001 mthmpps 35569 cnvnonrel 43577

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