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Definition df-rels 35761
Description: Define the relations class. Proper class relations (like I, see reli 5674) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 35763).

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 35763. 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 35786), 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 35761 (see df-refrels 35787 and the resulting dfrefrels2 35789 and dfrefrels3 35790).

3. Finally, in order to be able to work with proper classes (like iprc 7596) as well, we define the predicate of the relation (see df-refrel 35788) so that it is true for the relevant proper classes (see refrelid 35797), and that the element of the class of the required relations (e.g. elrefrels3 35794) and this predicate are the same in case of sets (see elrefrelsrel 35795). (Contributed by Peter Mazsa, 13-Jun-2018.)

Assertion
Ref Expression
df-rels Rels = 𝒫 (V × V)

Detailed syntax breakdown of Definition df-rels
StepHypRef Expression
1 crels 35491 . 2 class Rels
2 cvv 3473 . . . 4 class V
32, 2cxp 5529 . . 3 class (V × V)
43cpw 4515 . 2 class 𝒫 (V × V)
51, 4wceq 1537 1 wff Rels = 𝒫 (V × V)
Colors of variables: wff setvar class
This definition is referenced by:  elrels2  35762
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