Definition df-disjs 39479

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 39488).

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 39514. Alternate definitions are dfdisjs 39483, ... , dfdisjs5 39487. (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 38908	. 2 class Disjs
2		cdisjss 38907	. . 3 class Disjss
3		crels 38875	. . 3 class Rels
4	2, 3	cin 3907	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1570	1 wff Disjs = ( Disjss ∩ Rels )

This definition is used by:  dfdisjs  39483  eldisjsim2  39625