Theorem dfdisjALTV2

Description: Alternate definition of the disjoint relation predicate, cf. dffunALTV2 . (Contributed by Peter Mazsa, 27-Jul-2021)

	Ref	Expression
Assertion	dfdisjALTV2	⊢ ( Disj 𝑅 ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		df-disjALTV	⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡ 𝑅 ∧ Rel 𝑅 ) )
2		cnvrefrelcoss2	⊢ ( CnvRefRel ≀ ◡ 𝑅 ↔ ≀ ◡ 𝑅 ⊆ I )
3	2	anbi1i	⊢ ( ( CnvRefRel ≀ ◡ 𝑅 ∧ Rel 𝑅 ) ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) )
4	1 3	bitri	⊢ ( Disj 𝑅 ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) )