Definition df-rels 39089

Description: Define the relations class. Proper class relations (like I, see reli 5813) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 39091).

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 39091. 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 39239), 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 39089 (see df-refrels 39240 and the resulting dfrefrels2 39242 and dfrefrels3 39243).

3. Finally, in order to be able to work with proper classes (like iprc 7904) as well, we define the predicate of the relation (see df-refrel 39241) so that it is true for the relevant proper classes (see refrelid 39251), and that the element of the class of the required relations (e.g. elrefrels3 39248) and this predicate are the same in case of sets (see elrefrelsrel 39249). (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 38834	. 2 class Rels
2		cvv 3455	. . . 4 class V
3	2, 2	cxp 5659	. . 3 class (V × V)
4	3	cpw 4562	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1570	1 wff Rels = 𝒫 (V × V)

This definition is referenced by:  elrels2  39090