Theorem elsymrels4 39238

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 39230	. 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟}
2		cnveq 5863	. . 3 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
3		id 23	. . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
4	2, 3	eqeq12d 2786	. 2 ⊢ (𝑟 = 𝑅 → (◡𝑟 = 𝑟 ↔ ◡𝑅 = 𝑅))
5	1, 4	rabeqel 38856	1 ⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 = 𝑅 ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2150  ^◡ccnv 5664   Rels crels 38784   SymRels csymrels 38793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-rels 39039  df-ssr 39177  df-syms 39221  df-symrels 39222

This theorem is referenced by: (None)