Theorem elsymrels5 38557

Description: Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)

Assertion

Ref	Expression
elsymrels5	⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elsymrels5

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsymrels5 38549	. 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
2		breq 5145	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
3		breq 5145	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑥 ↔ 𝑦𝑅𝑥))
4	2, 3	bibi12d 345	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥) ↔ (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))
5	4	2albidv 1923	. 2 ⊢ (𝑟 = 𝑅 → (∀𝑥∀𝑦(𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))
6	1, 5	rabeqel 38255	1 ⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108   class class class wbr 5143   Rels crels 38184   SymRels csymrels 38193

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-rels 38486  df-ssr 38499  df-syms 38543  df-symrels 38544

This theorem is referenced by: (None)