Theorem elrelscnveq2 39752

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 6119	. . . 4 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
2	1	a1i 11	. . 3 ⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
3		elrelsrelim 39566	. . . . . . 7 ⊢ (𝑅 ∈ Rels → Rel 𝑅)
4		dfrel2 6192	. . . . . . 7 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
5	3, 4	sylib 221	. . . . . 6 ⊢ (𝑅 ∈ Rels → ◡◡𝑅 = 𝑅)
6	5	sseq1d 3944	. . . . 5 ⊢ (𝑅 ∈ Rels → (◡◡𝑅 ⊆ ◡𝑅 ↔ 𝑅 ⊆ ◡𝑅))
7		cnvsym 6119	. . . . 5 ⊢ (◡◡𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥))
8	6, 7	bitr3di 289	. . . 4 ⊢ (𝑅 ∈ Rels → (𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥)))
9		relbrcnvg 6112	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
10	3, 9	syl 18	. . . . . 6 ⊢ (𝑅 ∈ Rels → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
11		relbrcnvg 6112	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
12	3, 11	syl 18	. . . . . 6 ⊢ (𝑅 ∈ Rels → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
13	10, 12	imbi12d 347	. . . . 5 ⊢ (𝑅 ∈ Rels → ((𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥) ↔ (𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
14	13	2albidv 1958	. . . 4 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥◡𝑅𝑦 → 𝑦◡𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
15	8, 14	bitrd 282	. . 3 ⊢ (𝑅 ∈ Rels → (𝑅 ⊆ ◡𝑅 ↔ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
16	2, 15	anbi12d 644	. 2 ⊢ (𝑅 ∈ Rels → ((◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ◡𝑅) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦))))
17		eqss 3928	. 2 ⊢ (◡𝑅 = 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ⊆ ◡𝑅))
18		2albiim 1925	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦(𝑦𝑅𝑥 → 𝑥𝑅𝑦)))
19	16, 17, 18	3bitr4g 317	1 ⊢ (𝑅 ∈ Rels → (◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   class class class wbr 5106  ^◡ccnv 5654  Rel wrel 5660   Rels crels 39308

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-rels 39563

This theorem is used by:  elrelscnveq4  39753  dfsymrels5  39755