Definition df-rels 38449

Description: Define the relations class. Proper class relations (like I, see reli 5780) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 38451).

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 38451. 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 38474), 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 38449 (see df-refrels 38475 and the resulting dfrefrels2 38477 and dfrefrels3 38478).

3. Finally, in order to be able to work with proper classes (like iprc 7867) as well, we define the predicate of the relation (see df-refrel 38476) so that it is true for the relevant proper classes (see refrelid 38486), and that the element of the class of the required relations (e.g. elrefrels3 38483) and this predicate are the same in case of sets (see elrefrelsrel 38484). (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 38144	. 2 class Rels
2		cvv 3444	. . . 4 class V
3	2, 2	cxp 5629	. . 3 class (V × V)
4	3	cpw 4559	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1540	1 wff Rels = 𝒫 (V × V)

This definition is referenced by:  elrels2  38450