Definition df-rels 35885

Description: Define the relations class. Proper class relations (like I, see reli 5662) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 35887).

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 35887. 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 35910), 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 35885 (see df-refrels 35911 and the resulting dfrefrels2 35913 and dfrefrels3 35914).

3. Finally, in order to be able to work with proper classes (like iprc 7600) as well, we define the predicate of the relation (see df-refrel 35912) so that it is true for the relevant proper classes (see refrelid 35921), and that the element of the class of the required relations (e.g. elrefrels3 35918) and this predicate are the same in case of sets (see elrefrelsrel 35919). (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 35615	. 2 class Rels
2		cvv 3441	. . . 4 class V
3	2, 2	cxp 5517	. . 3 class (V × V)
4	3	cpw 4497	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1538	1 wff Rels = 𝒫 (V × V)

This definition is referenced by:  elrels2  35886