Definition df-rels 35740

Description: Define the relations class. Proper class relations (like I, see reli 5698) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 35742).

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 35742. 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 35765), 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 35740 (see df-refrels 35766 and the resulting dfrefrels2 35768 and dfrefrels3 35769).

3. Finally, in order to be able to work with proper classes (like iprc 7618) as well, we define the predicate of the relation (see df-refrel 35767) so that it is true for the relevant proper classes (see refrelid 35776), and that the element of the class of the required relations (e.g. elrefrels3 35773) and this predicate are the same in case of sets (see elrefrelsrel 35774). (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 35470	. 2 class Rels
2		cvv 3494	. . . 4 class V
3	2, 2	cxp 5553	. . 3 class (V × V)
4	3	cpw 4539	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1537	1 wff Rels = 𝒫 (V × V)

This definition is referenced by:  elrels2  35741