Theorem cnvdif 4945

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 4917	. 2 ⊢ Rel ◡(𝐴 ∖ 𝐵)
2		difss 3202	. . 3 ⊢ (◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴
3		relcnv 4917	. . 3 ⊢ Rel ◡𝐴
4		relss 4626	. . 3 ⊢ ((◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 → (Rel ◡𝐴 → Rel (◡𝐴 ∖ ◡𝐵)))
5	2, 3, 4	mp2 16	. 2 ⊢ Rel (◡𝐴 ∖ ◡𝐵)
6		eldif 3080	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7		vex 2689	. . . 4 ⊢ 𝑥 ∈ V
8		vex 2689	. . . 4 ⊢ 𝑦 ∈ V
9	7, 8	opelcnv 4721	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∖ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵))
10		eldif 3080	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵))
11	7, 8	opelcnv 4721	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
12	7, 8	opelcnv 4721	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
13	12	notbii 657	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
14	11, 13	anbi12i 455	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
15	10, 14	bitri 183	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
16	6, 9, 15	3bitr4i 211	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∖ 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵))
17	1, 5, 16	eqrelriiv 4633	1 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ∖ cdif 3068   ⊆ wss 3071  ⟨cop 3530  ^◡ccnv 4538  Rel wrel 4544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547

This theorem is referenced by: (None)