Theorem cnvdif 6036

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 6001	. 2 ⊢ Rel ◡(𝐴 ∖ 𝐵)
2		difss 4062	. . 3 ⊢ (◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴
3		relcnv 6001	. . 3 ⊢ Rel ◡𝐴
4		relss 5682	. . 3 ⊢ ((◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 → (Rel ◡𝐴 → Rel (◡𝐴 ∖ ◡𝐵)))
5	2, 3, 4	mp2 9	. 2 ⊢ Rel (◡𝐴 ∖ ◡𝐵)
6		eldif 3893	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7		vex 3426	. . . 4 ⊢ 𝑥 ∈ V
8		vex 3426	. . . 4 ⊢ 𝑦 ∈ V
9	7, 8	opelcnv 5779	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∖ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵))
10		eldif 3893	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵))
11	7, 8	opelcnv 5779	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
12	7, 8	opelcnv 5779	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
13	12	notbii 319	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
14	11, 13	anbi12i 626	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
15	10, 14	bitri 274	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
16	6, 9, 15	3bitr4i 302	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∖ 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵))
17	1, 5, 16	eqrelriiv 5689	1 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880   ⊆ wss 3883  ⟨cop 4564  ^◡ccnv 5579  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588

This theorem is referenced by:  cnvin  6037  gtiso  30935  gsumhashmul  31218  mthmpps  33444  cnvnonrel  41085