Theorem elsymrels4 35824

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 35816	. 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟}
2		cnveq 5737	. . 3 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
3		id 22	. . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
4	2, 3	eqeq12d 2836	. 2 ⊢ (𝑟 = 𝑅 → (◡𝑟 = 𝑟 ↔ ◡𝑅 = 𝑅))
5	1, 4	rabeqel 35549	1 ⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 = 𝑅 ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ^◡ccnv 5547   Rels crels 35488   SymRels csymrels 35497

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-opab 5122  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-rels 35758  df-ssr 35771  df-syms 35811  df-symrels 35812

This theorem is referenced by: (None)