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Theorem elrelscnveq2 38475
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 6135 . . . 4 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
21a1i 11 . . 3 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
3 elrelsrelim 38470 . . . . . . 7 (𝑅 ∈ Rels → Rel 𝑅)
4 dfrel2 6211 . . . . . . 7 (Rel 𝑅𝑅 = 𝑅)
53, 4sylib 218 . . . . . 6 (𝑅 ∈ Rels → 𝑅 = 𝑅)
65sseq1d 4027 . . . . 5 (𝑅 ∈ Rels → (𝑅𝑅𝑅𝑅))
7 cnvsym 6135 . . . . 5 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
86, 7bitr3di 286 . . . 4 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
9 relbrcnvg 6126 . . . . . . 7 (Rel 𝑅 → (𝑥𝑅𝑦𝑦𝑅𝑥))
103, 9syl 17 . . . . . 6 (𝑅 ∈ Rels → (𝑥𝑅𝑦𝑦𝑅𝑥))
11 relbrcnvg 6126 . . . . . . 7 (Rel 𝑅 → (𝑦𝑅𝑥𝑥𝑅𝑦))
123, 11syl 17 . . . . . 6 (𝑅 ∈ Rels → (𝑦𝑅𝑥𝑥𝑅𝑦))
1310, 12imbi12d 344 . . . . 5 (𝑅 ∈ Rels → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥𝑥𝑅𝑦)))
14132albidv 1921 . . . 4 (𝑅 ∈ Rels → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
158, 14bitrd 279 . . 3 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
162, 15anbi12d 632 . 2 (𝑅 ∈ Rels → ((𝑅𝑅𝑅𝑅) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦))))
17 eqss 4011 . 2 (𝑅 = 𝑅 ↔ (𝑅𝑅𝑅𝑅))
18 2albiim 1888 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
1916, 17, 183bitr4g 314 1 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  wss 3963   class class class wbr 5148  ccnv 5688  Rel wrel 5694   Rels crels 38164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-rels 38467
This theorem is referenced by:  elrelscnveq4  38476  dfsymrels5  38530
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