Theorem elsymrels4 39316

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

Assertion

Ref	Expression
elsymrels4	⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 = 𝑅 ∧ 𝑅 ∈ Rels ))

Proof of Theorem elsymrels4

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsymrels4 39308	. 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟}
2		cnveq 5858	. . 3 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
3		id 23	. . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
4	2, 3	eqeq12d 2778	. 2 ⊢ (𝑟 = 𝑅 → (◡𝑟 = 𝑟 ↔ ◡𝑅 = 𝑅))
5	1, 4	rabeqel 38934	1 ⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 = 𝑅 ∧ 𝑅 ∈ Rels ))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ^◡ccnv 5659   Rels crels 38862   SymRels csymrels 38871

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-dm 5670  df-rn 5671  df-res 5672  df-rels 39117  df-ssr 39255  df-syms 39299  df-symrels 39300

This theorem is used by: (None)