Definition df-rels 35719

Description: Define the relations class. Proper class relations (like I, see reli 5692) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 35721).

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 35721. 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 35744), 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 35719 (see df-refrels 35745 and the resulting dfrefrels2 35747 and dfrefrels3 35748).

3. Finally, in order to be able to work with proper classes (like iprc 7612) as well, we define the predicate of the relation (see df-refrel 35746) so that it is true for the relevant proper classes (see refrelid 35755), and that the element of the class of the required relations (e.g. elrefrels3 35752) and this predicate are the same in case of sets (see elrefrelsrel 35753). (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 35449	. 2 class Rels
2		cvv 3494	. . . 4 class V
3	2, 2	cxp 5547	. . 3 class (V × V)
4	3	cpw 4538	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1533	1 wff Rels = 𝒫 (V × V)

This definition is referenced by:  elrels2  35720