Definition df-disjs 39124

Description: Define the disjoint relations class, i.e., the class of disjoints. We need Disjs for the definition of Parts and Part for the Partition-Equivalence Theorems: this need for Parts as disjoint relations on their domain quotients is the reason why we must define Disjs instead of simply using converse functions (cf. dfdisjALTV 39133).

The element of the class of disjoints and the disjoint predicate are the same, that is (𝑅 ∈ Disjs ↔ Disj 𝑅) when 𝑅 is a set, see eldisjsdisj 39159. Alternate definitions are dfdisjs 39128, ... , dfdisjs5 39132. (Contributed by Peter Mazsa, 17-Jul-2021.)

Assertion

Ref	Expression
df-disjs	⊢ Disjs = ( Disjss ∩ Rels )

Detailed syntax breakdown of Definition df-disjs

Step	Hyp	Ref	Expression
1		cdisjs 38553	. 2 class Disjs
2		cdisjss 38552	. . 3 class Disjss
3		crels 38520	. . 3 class Rels
4	2, 3	cin 3889	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1542	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  39128  eldisjsim2  39270