Theorem cnvdif 5072

Description: Distributive law for converse over set 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 5043	. 2 ⊢ Rel ◡(𝐴 ∖ 𝐵)
2		difss 3285	. . 3 ⊢ (◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴
3		relcnv 5043	. . 3 ⊢ Rel ◡𝐴
4		relss 4746	. . 3 ⊢ ((◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 → (Rel ◡𝐴 → Rel (◡𝐴 ∖ ◡𝐵)))
5	2, 3, 4	mp2 16	. 2 ⊢ Rel (◡𝐴 ∖ ◡𝐵)
6		eldif 3162	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7		vex 2763	. . . 4 ⊢ 𝑥 ∈ V
8		vex 2763	. . . 4 ⊢ 𝑦 ∈ V
9	7, 8	opelcnv 4844	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∖ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵))
10		eldif 3162	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵))
11	7, 8	opelcnv 4844	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
12	7, 8	opelcnv 4844	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
13	12	notbii 669	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
14	11, 13	anbi12i 460	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
15	10, 14	bitri 184	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
16	6, 9, 15	3bitr4i 212	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∖ 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵))
17	1, 5, 16	eqrelriiv 4753	1 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1364   ∈ wcel 2164   ∖ cdif 3150   ⊆ wss 3153  ⟨cop 3621  ^◡ccnv 4658  Rel wrel 4664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667

This theorem is referenced by: (None)