Theorem cnvsymOLDOLD 6034

Description: Obsolete proof of cnvsym 6032 as of 23-Dec-2024. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cnvsymOLDOLD	⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cnvsymOLDOLD

Step	Hyp	Ref	Expression
1		alcom 2154	. 2 ⊢ (∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
2		relcnv 6022	. . 3 ⊢ Rel ◡𝑅
3		ssrel 5704	. . 3 ⊢ (Rel ◡𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅)))
4	2, 3	ax-mp 5	. 2 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
5		vex 3441	. . . . . 6 ⊢ 𝑦 ∈ V
6		vex 3441	. . . . . 6 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 5804	. . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
8		df-br 5082	. . . . 5 ⊢ (𝑦◡𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
9	7, 8	bitr3i 277	. . . 4 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ ◡𝑅)
10		df-br 5082	. . . 4 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
11	9, 10	imbi12i 351	. . 3 ⊢ ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ (⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
12	11	2albii 1820	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(⟨𝑦, 𝑥⟩ ∈ ◡𝑅 → ⟨𝑦, 𝑥⟩ ∈ 𝑅))
13	1, 4, 12	3bitr4i 303	1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   ∈ wcel 2104   ⊆ wss 3892  ⟨cop 4571   class class class wbr 5081  ^◡ccnv 5599  Rel wrel 5605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-xp 5606  df-rel 5607  df-cnv 5608

This theorem is referenced by: (None)