Theorem dfdisjALTV3

Description: Alternate definition of the disjoint relation predicate, cf. dffunALTV3 . (Contributed by Peter Mazsa, 28-Jul-2021)

	Ref	Expression
Assertion	dfdisjALTV3	⊢ ( Disj 𝑅 ↔ ( ∀ 𝑢 ∀ 𝑣 ∀ 𝑥 ( ( 𝑢 𝑅 𝑥 ∧ 𝑣 𝑅 𝑥 ) → 𝑢 = 𝑣 ) ∧ Rel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		dfdisjALTV2	⊢ ( Disj 𝑅 ↔ ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) )
2		cosscnvssid3	⊢ ( ≀ ◡ 𝑅 ⊆ I ↔ ∀ 𝑢 ∀ 𝑣 ∀ 𝑥 ( ( 𝑢 𝑅 𝑥 ∧ 𝑣 𝑅 𝑥 ) → 𝑢 = 𝑣 ) )
3	2	anbi1i	⊢ ( ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) ↔ ( ∀ 𝑢 ∀ 𝑣 ∀ 𝑥 ( ( 𝑢 𝑅 𝑥 ∧ 𝑣 𝑅 𝑥 ) → 𝑢 = 𝑣 ) ∧ Rel 𝑅 ) )
4	1 3	bitri	⊢ ( Disj 𝑅 ↔ ( ∀ 𝑢 ∀ 𝑣 ∀ 𝑥 ( ( 𝑢 𝑅 𝑥 ∧ 𝑣 𝑅 𝑥 ) → 𝑢 = 𝑣 ) ∧ Rel 𝑅 ) )