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Theorem cnvdif 6100

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 4087	. . 3 ⊢ (^◡𝐴 ∖ ^◡𝐵) ⊆ ^◡𝐴
3		relcnv 6062	. . 3 ⊢ Rel ^◡𝐴
4		relss 5730	. . 3 ⊢ ((^◡𝐴 ∖ ^◡𝐵) ⊆ ^◡𝐴 → (Rel ^◡𝐴 → Rel (^◡𝐴 ∖ ^◡𝐵)))
5	2, 3, 4	mp2 9	. 2 ⊢ Rel (^◡𝐴 ∖ ^◡𝐵)
6		eldif 3910	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7		vex 3443	. . . 4 ⊢ 𝑥 ∈ V
8		vex 3443	. . . 4 ⊢ 𝑦 ∈ V
9	7, 8	opelcnv 5829	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡(𝐴 ∖ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵))
10		eldif 3910	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (^◡𝐴 ∖ ^◡𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ^◡𝐵))
11	7, 8	opelcnv 5829	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
12	7, 8	opelcnv 5829	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
13	12	notbii 320	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ ^◡𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
14	11, 13	anbi12i 629	. . . 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 5738	1 ⊢ ^◡(𝐴 ∖ 𝐵) = (^◡𝐴 ∖ ^◡𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∖ cdif 3897 ⊆ wss 3900 ⟨cop 4585 ^◡ccnv 5622 Rel wrel 5628

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2707 ax-sep 5240 ax-nul 5250 ax-pr 5376

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2714 df-cleq 2727 df-clel 2810 df-rab 3399 df-v 3441 df-dif 3903 df-un 3905 df-ss 3917 df-nul 4285 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-br 5098 df-opab 5160 df-xp 5629 df-rel 5630 df-cnv 5631

This theorem is referenced by: cnvin 6101 gtiso 32759 gsumhashmul 33129 mthmpps 35755 cnvnonrel 43866

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