Definition df-rels 38522

Description: Define the relations class. Proper class relations (like I, see reli 5761) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 38524).

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 38524. 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 38547), 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 38522 (see df-refrels 38548 and the resulting dfrefrels2 38550 and dfrefrels3 38551).

3. Finally, in order to be able to work with proper classes (like iprc 7836) as well, we define the predicate of the relation (see df-refrel 38549) so that it is true for the relevant proper classes (see refrelid 38559), and that the element of the class of the required relations (e.g. elrefrels3 38556) and this predicate are the same in case of sets (see elrefrelsrel 38557). (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 38217	. 2 class Rels
2		cvv 3436	. . . 4 class V
3	2, 2	cxp 5609	. . 3 class (V × V)
4	3	cpw 4545	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1541	1 wff Rels = 𝒫 (V × V)

This definition is referenced by:  elrels2  38523