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Theorem elsymrels3 38536
Description: Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
Assertion
Ref Expression
elsymrels3 (𝑅 ∈ SymRels ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elsymrels3
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dfsymrels3 38528 . 2 SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}
2 breq 5150 . . . 4 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
3 breq 5150 . . . 4 (𝑟 = 𝑅 → (𝑦𝑟𝑥𝑦𝑅𝑥))
42, 3imbi12d 344 . . 3 (𝑟 = 𝑅 → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥)))
542albidv 1921 . 2 (𝑟 = 𝑅 → (∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
61, 5rabeqel 38236 1 (𝑅 ∈ SymRels ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106   class class class wbr 5148   Rels crels 38164   SymRels csymrels 38173
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-in 3970  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-dm 5699  df-rn 5700  df-res 5701  df-rels 38467  df-ssr 38480  df-syms 38524  df-symrels 38525
This theorem is referenced by: (None)
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