Definition df-syms 37412

Description: Define the class of all symmetric sets. It is used only by df-symrels 37413.

Note the similarity of Definitions df-refs 37380, df-syms 37412 and df-trs 37442, cf. the comment of dfrefrels2 37383. (Contributed by Peter Mazsa, 19-Jul-2019.)

Assertion

Ref	Expression
df-syms	⊢ Syms = {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

Detailed syntax breakdown of Definition df-syms

Step	Hyp	Ref	Expression
1		csyms 37053	. 2 class Syms
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1541	. . . . . 6 class 𝑥
4	3	cdm 5677	. . . . . . 7 class dom 𝑥
5	3	crn 5678	. . . . . . 7 class ran 𝑥
6	4, 5	cxp 5675	. . . . . 6 class (dom 𝑥 × ran 𝑥)
7	3, 6	cin 3948	. . . . 5 class (𝑥 ∩ (dom 𝑥 × ran 𝑥))
8	7	ccnv 5676	. . . 4 class ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥))
9		cssr 37046	. . . 4 class S
10	8, 7, 9	wbr 5149	. . 3 wff ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))
11	10, 2	cab 2710	. 2 class {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
12	1, 11	wceq 1542	1 wff Syms = {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

This definition is referenced by:  dfsymrels2  37415