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Theorem elrelscnveq2 37666
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
elrelscnveq2 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elrelscnveq2
StepHypRef Expression
1 cnvsym 6112 . . . 4 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
21a1i 11 . . 3 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
3 elrelsrelim 37661 . . . . . . 7 (𝑅 ∈ Rels → Rel 𝑅)
4 dfrel2 6187 . . . . . . 7 (Rel 𝑅𝑅 = 𝑅)
53, 4sylib 217 . . . . . 6 (𝑅 ∈ Rels → 𝑅 = 𝑅)
65sseq1d 4012 . . . . 5 (𝑅 ∈ Rels → (𝑅𝑅𝑅𝑅))
7 cnvsym 6112 . . . . 5 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
86, 7bitr3di 285 . . . 4 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
9 relbrcnvg 6103 . . . . . . 7 (Rel 𝑅 → (𝑥𝑅𝑦𝑦𝑅𝑥))
103, 9syl 17 . . . . . 6 (𝑅 ∈ Rels → (𝑥𝑅𝑦𝑦𝑅𝑥))
11 relbrcnvg 6103 . . . . . . 7 (Rel 𝑅 → (𝑦𝑅𝑥𝑥𝑅𝑦))
123, 11syl 17 . . . . . 6 (𝑅 ∈ Rels → (𝑦𝑅𝑥𝑥𝑅𝑦))
1310, 12imbi12d 343 . . . . 5 (𝑅 ∈ Rels → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥𝑥𝑅𝑦)))
14132albidv 1924 . . . 4 (𝑅 ∈ Rels → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
158, 14bitrd 278 . . 3 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
162, 15anbi12d 629 . 2 (𝑅 ∈ Rels → ((𝑅𝑅𝑅𝑅) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦))))
17 eqss 3996 . 2 (𝑅 = 𝑅 ↔ (𝑅𝑅𝑅𝑅))
18 2albiim 1891 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
1916, 17, 183bitr4g 313 1 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1537   = wceq 1539  wcel 2104  wss 3947   class class class wbr 5147  ccnv 5674  Rel wrel 5680   Rels crels 37348
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-rels 37658
This theorem is referenced by:  elrelscnveq4  37667  dfsymrels5  37721
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