Definition df-rels 37350

Description: Define the relations class. Proper class relations (like I, see reli 5826) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 37352).

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 37352. 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 37375), 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 37350 (see df-refrels 37376 and the resulting dfrefrels2 37378 and dfrefrels3 37379).

3. Finally, in order to be able to work with proper classes (like iprc 7903) as well, we define the predicate of the relation (see df-refrel 37377) so that it is true for the relevant proper classes (see refrelid 37387), and that the element of the class of the required relations (e.g. elrefrels3 37384) and this predicate are the same in case of sets (see elrefrelsrel 37385). (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 37040	. 2 class Rels
2		cvv 3474	. . . 4 class V
3	2, 2	cxp 5674	. . 3 class (V × V)
4	3	cpw 4602	. 2 class 𝒫 (V × V)
5	1, 4	wceq 1541	1 wff Rels = 𝒫 (V × V)

This definition is referenced by:  elrels2  37351