Definition df-rels 36612

Description: Define the relations class. Proper class relations (like I, see reli 5735) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 36614).

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 36614. 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 36637), 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 36612 (see df-refrels 36638 and the resulting dfrefrels2 36640 and dfrefrels3 36641).

3. Finally, in order to be able to work with proper classes (like iprc 7755) as well, we define the predicate of the relation (see df-refrel 36639) so that it is true for the relevant proper classes (see refrelid 36648), and that the element of the class of the required relations (e.g. elrefrels3 36645) and this predicate are the same in case of sets (see elrefrelsrel 36646). (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 36344	. 2 class Rels
2		cvv 3431	. . . 4 class V
3	2, 2	cxp 5588	. . 3 class (V × V)
4	3	cpw 4539	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1542	1 wff Rels = 𝒫 (V × V)

This definition is referenced by:  elrels2  36613