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Definition df-refs 34602
Description: Define the class of all reflexive sets. It is used only by df-refrels 34603. We use subset relation S (df-ssr 34590) here to be able to define converse reflexivity (df-cnvrefs 34615), cf. the comment of df-ssr 34590. The elements of this class are not necessarily relations (vs. df-refrels 34603).

Note the similarity of the definitions df-refs 34602, df-syms 34630 and ~? df-trs , cf. the comments of dfrefrels2 34605. (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
StepHypRef Expression
1 crefs 34319 . 2 class Refs
2 cid 5157 . . . . 5 class I
3 vx . . . . . . . 8 setvar 𝑥
43cv 1630 . . . . . . 7 class 𝑥
54cdm 5250 . . . . . 6 class dom 𝑥
64crn 5251 . . . . . 6 class ran 𝑥
75, 6cxp 5248 . . . . 5 class (dom 𝑥 × ran 𝑥)
82, 7cin 3722 . . . 4 class ( I ∩ (dom 𝑥 × ran 𝑥))
94, 7cin 3722 . . . 4 class (𝑥 ∩ (dom 𝑥 × ran 𝑥))
10 cssr 34318 . . . 4 class S
118, 9, 10wbr 4787 . . 3 wff ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))
1211, 3cab 2757 . 2 class {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
131, 12wceq 1631 1 wff Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Colors of variables: wff setvar class
This definition is referenced by:  dfrefrels2  34605
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