Definition df-rels 38625

Description: Define the relations class. Proper class relations (like I, see reli 5775) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 38627).

The class of relations is a great tool we can use when we define classes of different relations as nullary class constants as required by the 2. point in our Guidelines https://us.metamath.org/mpeuni/mathbox.html 38627. When we want to define a specific class of relations as a nullary class constant, the appropriate method is the following:

1. We define the specific nullary class constant for general sets (see e.g. df-refs 38763), then

2. we get the required class of relations by the intersection of the class of general sets above with the class of relations df-rels 38625 (see df-refrels 38764 and the resulting dfrefrels2 38766 and dfrefrels3 38767).

3. Finally, in order to be able to work with proper classes (like iprc 7853) as well, we define the predicate of the relation (see df-refrel 38765) so that it is true for the relevant proper classes (see refrelid 38775), and that the element of the class of the required relations (e.g. elrefrels3 38772) and this predicate are the same in case of sets (see elrefrelsrel 38773). (Contributed by Peter Mazsa, 13-Jun-2018.)

Assertion

Ref	Expression
df-rels	⊢ Rels = 𝒫 (V × V)

Detailed syntax breakdown of Definition df-rels

Step	Hyp	Ref	Expression
1		crels 38385	. 2 class Rels
2		cvv 3440	. . . 4 class V
3	2, 2	cxp 5622	. . 3 class (V × V)
4	3	cpw 4554	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1541	1 wff Rels = 𝒫 (V × V)

This definition is referenced by:  elrels2  38626