Definition df-syms 39271

Description: Define the class of all symmetric sets. It is used only by df-symrels 39272.

Note the similarity of Definitions df-refs 39239, df-syms 39271 and df-trs 39305, cf. the comment of dfrefrels2 39242. (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 38842	. 2 class Syms
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1569	. . . . . 6 class 𝑥
4	3	cdm 5661	. . . . . . 7 class dom 𝑥
5	3	crn 5662	. . . . . . 7 class ran 𝑥
6	4, 5	cxp 5659	. . . . . 6 class (dom 𝑥 × ran 𝑥)
7	3, 6	cin 3904	. . . . 5 class (𝑥 ∩ (dom 𝑥 × ran 𝑥))
8	7	ccnv 5660	. . . 4 class ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥))
9		cssr 38835	. . . 4 class S
10	8, 7, 9	wbr 5109	. . 3 wff ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))
11	10, 2	cab 2741	. 2 class {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
12	1, 11	wceq 1570	1 wff Syms = {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

This definition is referenced by:  dfsymrels2  39274