Theorem cnvdif 4730

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 4703	. 2 ⊢ Rel ◡(𝐴 ∖ 𝐵)
2		difss 3070	. . 3 ⊢ (◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴
3		relcnv 4703	. . 3 ⊢ Rel ◡𝐴
4		relss 4427	. . 3 ⊢ ((◡𝐴 ∖ ◡𝐵) ⊆ ◡𝐴 → (Rel ◡𝐴 → Rel (◡𝐴 ∖ ◡𝐵)))
5	2, 3, 4	mp2 16	. 2 ⊢ Rel (◡𝐴 ∖ ◡𝐵)
6		eldif 2927	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
7		vex 2560	. . . 4 ⊢ 𝑥 ∈ V
8		vex 2560	. . . 4 ⊢ 𝑦 ∈ V
9	7, 8	opelcnv 4517	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∖ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∖ 𝐵))
10		eldif 2927	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵))
11	7, 8	opelcnv 4517	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
12	7, 8	opelcnv 4517	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
13	12	notbii 594	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
14	11, 13	anbi12i 433	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ◡𝐴 ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
15	10, 14	bitri 173	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐴 ∧ ¬ ⟨𝑦, 𝑥⟩ ∈ 𝐵))
16	6, 9, 15	3bitr4i 201	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∖ 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∖ ◡𝐵))
17	1, 5, 16	eqrelriiv 4434	1 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ∖ cdif 2914   ⊆ wss 2917  ⟨cop 3378  ^◡ccnv 4344  Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353

This theorem is referenced by: (None)