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Theorem elsymrels5 37047
Description: Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
Assertion
Ref Expression
elsymrels5 (𝑅 ∈ SymRels ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elsymrels5
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dfsymrels5 37039 . 2 SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}
2 breq 5112 . . . 4 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
3 breq 5112 . . . 4 (𝑟 = 𝑅 → (𝑦𝑟𝑥𝑦𝑅𝑥))
42, 3bibi12d 346 . . 3 (𝑟 = 𝑅 → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥)))
542albidv 1927 . 2 (𝑟 = 𝑅 → (∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
61, 5rabeqel 36743 1 (𝑅 ∈ SymRels ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107   class class class wbr 5110   Rels crels 36665   SymRels csymrels 36674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-rels 36976  df-ssr 36989  df-syms 37033  df-symrels 37034
This theorem is referenced by: (None)
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