Definition df-rels 39638

Description: Define the relations class. Proper class relations (like I, see reli 5814) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 39640).

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 39640. 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 39788), 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 39638 (see df-refrels 39789 and the resulting dfrefrels2 39791 and dfrefrels3 39792).

3. Finally, in order to be able to work with proper classes (like iprc 7957) as well, we define the predicate of the relation (see df-refrel 39790) so that it is true for the relevant proper classes (see refrelid 39800), and that the element of the class of the required relations (e.g. elrefrels3 39797) and this predicate are the same in case of sets (see elrefrelsrel 39798). (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 39383	. 2 class Rels
2		cvv 3454	. . . 4 class V
3	2, 2	cxp 5656	. . 3 class (V × V)
4	3	cpw 4560	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1570	1 wff Rels = 𝒫 (V × V)

This definition is used by:  elrels2  39639