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Theorem elrelscnveq2 36353
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 5984 . . . 4 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
21a1i 11 . . 3 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
3 elrelsrelim 36348 . . . . . . 7 (𝑅 ∈ Rels → Rel 𝑅)
4 dfrel2 6057 . . . . . . 7 (Rel 𝑅𝑅 = 𝑅)
53, 4sylib 221 . . . . . 6 (𝑅 ∈ Rels → 𝑅 = 𝑅)
65sseq1d 3937 . . . . 5 (𝑅 ∈ Rels → (𝑅𝑅𝑅𝑅))
7 cnvsym 5984 . . . . 5 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
86, 7bitr3di 289 . . . 4 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
9 relbrcnvg 5978 . . . . . . 7 (Rel 𝑅 → (𝑥𝑅𝑦𝑦𝑅𝑥))
103, 9syl 17 . . . . . 6 (𝑅 ∈ Rels → (𝑥𝑅𝑦𝑦𝑅𝑥))
11 relbrcnvg 5978 . . . . . . 7 (Rel 𝑅 → (𝑦𝑅𝑥𝑥𝑅𝑦))
123, 11syl 17 . . . . . 6 (𝑅 ∈ Rels → (𝑦𝑅𝑥𝑥𝑅𝑦))
1310, 12imbi12d 348 . . . . 5 (𝑅 ∈ Rels → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥𝑥𝑅𝑦)))
14132albidv 1931 . . . 4 (𝑅 ∈ Rels → (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
158, 14bitrd 282 . . 3 (𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
162, 15anbi12d 634 . 2 (𝑅 ∈ Rels → ((𝑅𝑅𝑅𝑅) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦))))
17 eqss 3921 . 2 (𝑅 = 𝑅 ↔ (𝑅𝑅𝑅𝑅))
18 2albiim 1898 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦(𝑦𝑅𝑥𝑥𝑅𝑦)))
1916, 17, 183bitr4g 317 1 (𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1541   = wceq 1543  wcel 2110  wss 3871   class class class wbr 5058  ccnv 5555  Rel wrel 5561   Rels crels 36077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pr 5327
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-op 4553  df-br 5059  df-opab 5121  df-xp 5562  df-rel 5563  df-cnv 5564  df-rels 36345
This theorem is referenced by:  elrelscnveq4  36354  dfsymrels5  36404
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