Theorem elrelscnveq2 37028

Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)

Assertion

Ref	Expression
elrelscnveq2	⊢ (𝑅 ∈ Rels → (◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elrelscnveq2

Step	Hyp	Ref	Expression
1		cnvsym 6071	. . . 4 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
2	1	a1i 11	. . 3 ⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
3		elrelsrelim 37023	. . . . . . 7 ⊢ (𝑅 ∈ Rels → Rel 𝑅)
4		dfrel2 6146	. . . . . . 7 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
5	3, 4	sylib 217	. . . . . 6 ⊢ (𝑅 ∈ Rels → ◡◡𝑅 = 𝑅)
6	5	sseq1d 3978	. . . . 5 ⊢ (𝑅 ∈ Rels → (◡◡𝑅 ⊆ ◡𝑅 ↔ 𝑅 ⊆ ◡𝑅))
7		cnvsym 6071	. . . . 5 ⊢ (◡◡𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥))
8	6, 7	bitr3di 285	. . . 4 ⊢ (𝑅 ∈ Rels → (𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥)))
9		relbrcnvg 6062	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
10	3, 9	syl 17	. . . . . 6 ⊢ (𝑅 ∈ Rels → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
11		relbrcnvg 6062	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
12	3, 11	syl 17	. . . . . 6 ⊢ (𝑅 ∈ Rels → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
13	10, 12	imbi12d 344	. . . . 5 ⊢ (𝑅 ∈ Rels → ((𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥) ↔ (𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
14	13	2albidv 1926	. . . 4 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
15	8, 14	bitrd 278	. . 3 ⊢ (𝑅 ∈ Rels → (𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
16	2, 15	anbi12d 631	. 2 ⊢ (𝑅 ∈ Rels → ((◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ◡𝑅) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦))))
17		eqss 3962	. 2 ⊢ (◡𝑅 = 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ◡𝑅))
18		2albiim 1893	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
19	16, 17, 18	3bitr4g 313	1 ⊢ (𝑅 ∈ Rels → (◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106   ⊆ wss 3913   class class class wbr 5110  ^◡ccnv 5637  Rel wrel 5643   Rels crels 36709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-rels 37020

This theorem is referenced by:  elrelscnveq4  37029  dfsymrels5  37083