Theorem dfdisjALTV

Description: Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs why we need disjoint relations instead of converse functions anyway. (Contributed by Peter Mazsa, 27-Jul-2021)

	Ref	Expression
Assertion	dfdisjALTV	⊢ ( Disj 𝑅 ↔ ( FunALTV ◡ 𝑅 ∧ Rel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		df-disjALTV	⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡ 𝑅 ∧ Rel 𝑅 ) )
2		relcnv	⊢ Rel ◡ 𝑅
3		df-funALTV	⊢ ( FunALTV ◡ 𝑅 ↔ ( CnvRefRel ≀ ◡ 𝑅 ∧ Rel ◡ 𝑅 ) )
4	2 3	mpbiran2	⊢ ( FunALTV ◡ 𝑅 ↔ CnvRefRel ≀ ◡ 𝑅 )
5	4	anbi1i	⊢ ( ( FunALTV ◡ 𝑅 ∧ Rel 𝑅 ) ↔ ( CnvRefRel ≀ ◡ 𝑅 ∧ Rel 𝑅 ) )
6	1 5	bitr4i	⊢ ( Disj 𝑅 ↔ ( FunALTV ◡ 𝑅 ∧ Rel 𝑅 ) )