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Theorem cnvdif 6175

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 6134	. 2 ⊢ Rel ^◡(𝐴 ∖ 𝐵)
2		difss 4159	. . 3 ⊢ (^◡𝐴 ∖ ^◡𝐵) ⊆ ^◡𝐴
3		relcnv 6134	. . 3 ⊢ Rel ^◡𝐴
4		relss 5805	. . 3 ⊢ ((^◡𝐴 ∖ ^◡𝐵) ⊆ ^◡𝐴 → (Rel ^◡𝐴 → Rel (^◡𝐴 ∖ ^◡𝐵)))
5	2, 3, 4	mp2 9	. 2 ⊢ Rel (^◡𝐴 ∖ ^◡𝐵)
6		eldif 3986	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7		vex 3492	. . . 4 ⊢ 𝑥 ∈ V
8		vex 3492	. . . 4 ⊢ 𝑦 ∈ V
9	7, 8	opelcnv 5906	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡(𝐴 ∖ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵))
10		eldif 3986	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (^◡𝐴 ∖ ^◡𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ^◡𝐵))
11	7, 8	opelcnv 5906	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
12	7, 8	opelcnv 5906	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
13	12	notbii 320	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ ^◡𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
14	11, 13	anbi12i 627	. . . 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 5814	1 ⊢ ^◡(𝐴 ∖ 𝐵) = (^◡𝐴 ∖ ^◡𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∖ cdif 3973 ⊆ wss 3976 ⟨cop 4654 ^◡ccnv 5699 Rel wrel 5705

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-xp 5706 df-rel 5707 df-cnv 5708

This theorem is referenced by: cnvin 6176 gtiso 32712 gsumhashmul 33040 mthmpps 35550 cnvnonrel 43550

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