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Theorem elrelscnveq3 35611
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 3979 . 2 (𝑅 = 𝑅 ↔ (𝑅𝑅𝑅𝑅))
2 cnvsym 5967 . . . . . . 7 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
32biimpi 217 . . . . . 6 (𝑅𝑅 → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
43a1d 25 . . . . 5 (𝑅𝑅 → (𝑅 ∈ Rels → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
54adantl 482 . . . 4 ((𝑅𝑅𝑅𝑅) → (𝑅 ∈ Rels → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
65com12 32 . . 3 (𝑅 ∈ Rels → ((𝑅𝑅𝑅𝑅) → ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
7 elrelsrelim 35608 . . . . . 6 (𝑅 ∈ Rels → Rel 𝑅)
8 dfrel2 6039 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
97, 8sylib 219 . . . . 5 (𝑅 ∈ Rels → 𝑅 = 𝑅)
10 cnvss 5736 . . . . . . 7 (𝑅𝑅𝑅𝑅)
11 sseq1 3989 . . . . . . 7 (𝑅 = 𝑅 → (𝑅𝑅𝑅𝑅))
1210, 11syl5ibcom 246 . . . . . 6 (𝑅𝑅 → (𝑅 = 𝑅𝑅𝑅))
132, 12sylbir 236 . . . . 5 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (𝑅 = 𝑅𝑅𝑅))
149, 13syl5com 31 . . . 4 (𝑅 ∈ Rels → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅))
152biimpri 229 . . . 4 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑅𝑅)
1614, 15jca2 514 . . 3 (𝑅 ∈ Rels → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (𝑅𝑅𝑅𝑅)))
176, 16impbid 213 . 2 (𝑅 ∈ Rels → ((𝑅𝑅𝑅𝑅) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
181, 17syl5bb 284 1 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wcel 2105  wss 3933   class class class wbr 5057  ccnv 5547  Rel wrel 5553   Rels crels 35336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-rels 35605
This theorem is referenced by:  elrelscnveq  35612
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