Theorem cnvsymOLD 6074

Description: Obsolete version of cnvsym 6073 as of 29-Dec-2024. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cnvsymOLD

Step	Hyp	Ref	Expression
1		relcnv 6064	. . 3 ⊢ Rel ◡𝑅
2		ssrel3 5740	. . 3 ⊢ (Rel ◡𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥)))
3	1, 2	ax-mp 5	. 2 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥))
4		alcom 2160	. 2 ⊢ (∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥))
5		vex 3448	. . . . 5 ⊢ 𝑦 ∈ V
6		vex 3448	. . . . 5 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 5836	. . . 4 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
8	7	imbi1i 349	. . 3 ⊢ ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
9	8	2albii 1820	. 2 ⊢ (∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
10	3, 4, 9	3bitri 297	1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   ⊆ wss 3911   class class class wbr 5102  ^◡ccnv 5630  Rel wrel 5636

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639

This theorem is referenced by: (None)