Definition df-rels 39189

Description: Define the relations class. Proper class relations (like I, see reli 5807) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 39191).

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 39191. 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 39339), 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 39189 (see df-refrels 39340 and the resulting dfrefrels2 39342 and dfrefrels3 39343).

3. Finally, in order to be able to work with proper classes (like iprc 7909) as well, we define the predicate of the relation (see df-refrel 39341) so that it is true for the relevant proper classes (see refrelid 39351), and that the element of the class of the required relations (e.g. elrefrels3 39348) and this predicate are the same in case of sets (see elrefrelsrel 39349). (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 38934	. 2 class Rels
2		cvv 3450	. . . 4 class V
3	2, 2	cxp 5653	. . 3 class (V × V)
4	3	cpw 4557	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1570	1 wff Rels = 𝒫 (V × V)

This definition is used by:  elrels2  39190