Definition df-disjALTV 37575

Description: Define the disjoint relation predicate, i.e., the disjoint predicate. A disjoint relation is a converse function of the relation by dfdisjALTV 37583, see the comment of df-disjs 37574 why we need disjoint relations instead of converse functions anyway.

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 37597. Alternate definitions are dfdisjALTV 37583, ... , dfdisjALTV5 37587. (Contributed by Peter Mazsa, 17-Jul-2021.)

Assertion

Ref	Expression
df-disjALTV	⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))

Detailed syntax breakdown of Definition df-disjALTV

Step	Hyp	Ref	Expression
1		cR	. . 3 class 𝑅
2	1	wdisjALTV 37077	. 2 wff Disj 𝑅
3	1	ccnv 5676	. . . . 5 class ◡𝑅
4	3	ccoss 37043	. . . 4 class ≀ ◡𝑅
5	4	wcnvrefrel 37052	. . 3 wff CnvRefRel ≀ ◡𝑅
6	1	wrel 5682	. . 3 wff Rel 𝑅
7	5, 6	wa 397	. 2 wff ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅)
8	2, 7	wb 205	1 wff ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))

This definition is referenced by:  dfdisjALTV  37583  dfdisjALTV2  37584  eldisjsdisj  37597  disjrel  37600