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Theorem elrelscnveq3 38493
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
elrelscnveq3 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elrelscnveq3
StepHypRef Expression
1 eqss 3998 . 2 (𝑅 = 𝑅 ↔ (𝑅𝑅𝑅𝑅))
2 cnvsym 6131 . . . . . . 7 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
32biimpi 216 . . . . . 6 (𝑅𝑅 → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
43a1d 25 . . . . 5 (𝑅𝑅 → (𝑅 ∈ Rels → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
54adantl 481 . . . 4 ((𝑅𝑅𝑅𝑅) → (𝑅 ∈ Rels → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
65com12 32 . . 3 (𝑅 ∈ Rels → ((𝑅𝑅𝑅𝑅) → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
7 elrelsrelim 38490 . . . . . 6 (𝑅 ∈ Rels → Rel 𝑅)
8 dfrel2 6208 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
97, 8sylib 218 . . . . 5 (𝑅 ∈ Rels → 𝑅 = 𝑅)
10 cnvss 5882 . . . . . . 7 (𝑅𝑅𝑅𝑅)
11 sseq1 4008 . . . . . . 7 (𝑅 = 𝑅 → (𝑅𝑅𝑅𝑅))
1210, 11syl5ibcom 245 . . . . . 6 (𝑅𝑅 → (𝑅 = 𝑅𝑅𝑅))
132, 12sylbir 235 . . . . 5 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (𝑅 = 𝑅𝑅𝑅))
149, 13syl5com 31 . . . 4 (𝑅 ∈ Rels → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅))
152biimpri 228 . . . 4 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅)
1614, 15jca2 513 . . 3 (𝑅 ∈ Rels → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (𝑅𝑅𝑅𝑅)))
176, 16impbid 212 . 2 (𝑅 ∈ Rels → ((𝑅𝑅𝑅𝑅) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
181, 17bitrid 283 1 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wcel 2107  wss 3950   class class class wbr 5142  ccnv 5683  Rel wrel 5689   Rels crels 38185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-rels 38487
This theorem is referenced by:  elrelscnveq  38494
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