Theorem elsymrels3 39350

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

Assertion

Ref	Expression
elsymrels3	⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elsymrels3

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsymrels3 39338	. 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)}
2		breq 5113	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
3		breq 5113	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑥 ↔ 𝑦𝑅𝑥))
4	2, 3	imbi12d 347	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 → 𝑦𝑟𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
5	4	2albidv 1956	. 2 ⊢ (𝑟 = 𝑅 → (∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
6	1, 5	rabeqel 38969	1 ⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146   class class class wbr 5111   Rels crels 38897   SymRels csymrels 38906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-rels 39152  df-ssr 39290  df-syms 39334  df-symrels 39335

This theorem is used by: (None)