Definition df-rels 37951

Description: Define the relations class. Proper class relations (like I, see reli 5822) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 37953).

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 37953. 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 37976), 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 37951 (see df-refrels 37977 and the resulting dfrefrels2 37979 and dfrefrels3 37980).

3. Finally, in order to be able to work with proper classes (like iprc 7913) as well, we define the predicate of the relation (see df-refrel 37978) so that it is true for the relevant proper classes (see refrelid 37988), and that the element of the class of the required relations (e.g. elrefrels3 37985) and this predicate are the same in case of sets (see elrefrelsrel 37986). (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 37644	. 2 class Rels
2		cvv 3470	. . . 4 class V
3	2, 2	cxp 5670	. . 3 class (V × V)
4	3	cpw 4598	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1534	1 wff Rels = 𝒫 (V × V)

This definition is referenced by:  elrels2  37952