Definition df-refs 35765

Description: Define the class of all reflexive sets. It is used only by df-refrels 35766. We use subset relation S (df-ssr 35753) here to be able to define converse reflexivity (df-cnvrefs 35778), see also the comment of df-ssr 35753. The elements of this class are not necessarily relations (versus df-refrels 35766).

Note the similarity of the definitions df-refs 35765, df-syms 35793 and df-trs 35823, cf. comments of dfrefrels2 35768. (Contributed by Peter Mazsa, 19-Jul-2019.)

Assertion

Ref	Expression
df-refs	⊢ Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

Detailed syntax breakdown of Definition df-refs

Step	Hyp	Ref	Expression
1		crefs 35472	. 2 class Refs
2		cid 5459	. . . . 5 class I
3		vx	. . . . . . . 8 setvar 𝑥
4	3	cv 1536	. . . . . . 7 class 𝑥
5	4	cdm 5555	. . . . . 6 class dom 𝑥
6	4	crn 5556	. . . . . 6 class ran 𝑥
7	5, 6	cxp 5553	. . . . 5 class (dom 𝑥 × ran 𝑥)
8	2, 7	cin 3935	. . . 4 class ( I ∩ (dom 𝑥 × ran 𝑥))
9	4, 7	cin 3935	. . . 4 class (𝑥 ∩ (dom 𝑥 × ran 𝑥))
10		cssr 35471	. . . 4 class S
11	8, 9, 10	wbr 5066	. . 3 wff ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))
12	11, 3	cab 2799	. 2 class {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
13	1, 12	wceq 1537	1 wff Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

This definition is referenced by:  dfrefrels2  35768