Definition df-disjs 39814

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 39823).

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 39849. Alternate definitions are dfdisjs 39818, ... , dfdisjs5 39822. (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 39243	. 2 class Disjs
2		cdisjss 39242	. . 3 class Disjss
3		crels 39210	. . 3 class Rels
4	2, 3	cin 4089	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1569	1 wff Disjs = ( Disjss ∩ Rels )

This definition is used by:  dfdisjs  39818  eldisjsim2  39960