Theorem elsymrels4 39491

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 39483	. 2 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟}
2		cnveq 5847	. . 3 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
3		id 23	. . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
4	2, 3	eqeq12d 2776	. 2 ⊢ (𝑟 = 𝑅 → (◡𝑟 = 𝑟 ↔ ◡𝑅 = 𝑅))
5	1, 4	rabeqel 39109	1 ⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 = 𝑅 ∧ 𝑅 ∈ Rels ))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ^◡ccnv 5646   Rels crels 39037   SymRels csymrels 39046

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-rels 39292  df-ssr 39430  df-syms 39474  df-symrels 39475

This theorem is used by: (None)