Definition df-rels 38978

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 38980).

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 38980. 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 39128), 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 38978 (see df-refrels 39129 and the resulting dfrefrels2 39131 and dfrefrels3 39132).

3. Finally, in order to be able to work with proper classes (like iprc 7907) as well, we define the predicate of the relation (see df-refrel 39130) so that it is true for the relevant proper classes (see refrelid 39140), and that the element of the class of the required relations (e.g. elrefrels3 39137) and this predicate are the same in case of sets (see elrefrelsrel 39138). (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 38723	. 2 class Rels
2		cvv 3463	. . . 4 class V
3	2, 2	cxp 5660	. . 3 class (V × V)
4	3	cpw 4567	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1567	1 wff Rels = 𝒫 (V × V)

This definition is referenced by:  elrels2  38979