Theorem elrelscnveq3 39200

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

Assertion

Ref	Expression
elrelscnveq3	⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elrelscnveq3

Step	Hyp	Ref	Expression
1		eqss 3960	. 2 ⊢ (𝑅 = ◡𝑅 ↔ (𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅))
2		cnvsym 6115	. . . . . . 7 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
3	2	biimpi 219	. . . . . 6 ⊢ (◡𝑅 ⊆ 𝑅 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
4	3	a1d 26	. . . . 5 ⊢ (◡𝑅 ⊆ 𝑅 → (𝑅 ∈ Rels → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
5	4	adantl 486	. . . 4 ⊢ ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) → (𝑅 ∈ Rels → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
6	5	com12 33	. . 3 ⊢ (𝑅 ∈ Rels → ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
7		elrelsrelim 39016	. . . . . 6 ⊢ (𝑅 ∈ Rels → Rel 𝑅)
8		dfrel2 6188	. . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
9	7, 8	sylib 221	. . . . 5 ⊢ (𝑅 ∈ Rels → ◡◡𝑅 = 𝑅)
10		cnvss 5859	. . . . . . 7 ⊢ (◡𝑅 ⊆ 𝑅 → ◡◡𝑅 ⊆ ◡𝑅)
11		sseq1 3970	. . . . . . 7 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ⊆ ◡𝑅 ↔ 𝑅 ⊆ ◡𝑅))
12	10, 11	syl5ibcom 248	. . . . . 6 ⊢ (◡𝑅 ⊆ 𝑅 → (◡◡𝑅 = 𝑅 → 𝑅 ⊆ ◡𝑅))
13	2, 12	sylbir 238	. . . . 5 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (◡◡𝑅 = 𝑅 → 𝑅 ⊆ ◡𝑅))
14	9, 13	syl5com 32	. . . 4 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → 𝑅 ⊆ ◡𝑅))
15	2	biimpri 231	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → ◡𝑅 ⊆ 𝑅)
16	14, 15	jca2 522	. . 3 ⊢ (𝑅 ∈ Rels → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅)))
17	6, 16	impbid 215	. 2 ⊢ (𝑅 ∈ Rels → ((𝑅 ⊆ ◡𝑅 ∧ ◡𝑅 ⊆ 𝑅) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
18	1, 17	bitrid 286	1 ⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913   class class class wbr 5113  ^◡ccnv 5661  Rel wrel 5667   Rels crels 38758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-rels 39013

This theorem is referenced by:  elrelscnveq  39201