Theorem elrelscnveq2 38491

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 6088	. . . 4 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
2	1	a1i 11	. . 3 ⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
3		elrelsrelim 38486	. . . . . . 7 ⊢ (𝑅 ∈ Rels → Rel 𝑅)
4		dfrel2 6165	. . . . . . 7 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
5	3, 4	sylib 218	. . . . . 6 ⊢ (𝑅 ∈ Rels → ◡◡𝑅 = 𝑅)
6	5	sseq1d 3981	. . . . 5 ⊢ (𝑅 ∈ Rels → (◡◡𝑅 ⊆ ◡𝑅 ↔ 𝑅 ⊆ ◡𝑅))
7		cnvsym 6088	. . . . 5 ⊢ (◡◡𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥))
8	6, 7	bitr3di 286	. . . 4 ⊢ (𝑅 ∈ Rels → (𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥)))
9		relbrcnvg 6079	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
10	3, 9	syl 17	. . . . . 6 ⊢ (𝑅 ∈ Rels → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
11		relbrcnvg 6079	. . . . . . 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 1923	. . . 4 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
15	8, 14	bitrd 279	. . 3 ⊢ (𝑅 ∈ Rels → (𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
16	2, 15	anbi12d 632	. 2 ⊢ (𝑅 ∈ Rels → ((◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ◡𝑅) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦))))
17		eqss 3965	. 2 ⊢ (◡𝑅 = 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ◡𝑅))
18		2albiim 1890	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
19	16, 17, 18	3bitr4g 314	1 ⊢ (𝑅 ∈ Rels → (◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ⊆ wss 3917   class class class wbr 5110  ^◡ccnv 5640  Rel wrel 5646   Rels crels 38178

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-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-rels 38483

This theorem is referenced by:  elrelscnveq4  38492  dfsymrels5  38546