Theorem disjxrn

Description: Two ways of saying that a range Cartesian product is disjoint. (Contributed by Peter Mazsa, 17-Jun-2020) (Revised by Peter Mazsa, 21-Sep-2021)

	Ref	Expression
Assertion	disjxrn	⊢ ( Disj ( 𝑅 ⋉ 𝑆 ) ↔ ( ≀ ◡ 𝑅 ∩ ≀ ◡ 𝑆 ) ⊆ I )

Proof

Step	Hyp	Ref	Expression
1		xrnrel	⊢ Rel ( 𝑅 ⋉ 𝑆 )
2		dfdisjALTV2	⊢ ( Disj ( 𝑅 ⋉ 𝑆 ) ↔ ( ≀ ◡ ( 𝑅 ⋉ 𝑆 ) ⊆ I ∧ Rel ( 𝑅 ⋉ 𝑆 ) ) )
3	1 2	mpbiran2	⊢ ( Disj ( 𝑅 ⋉ 𝑆 ) ↔ ≀ ◡ ( 𝑅 ⋉ 𝑆 ) ⊆ I )
4		1cosscnvxrn	⊢ ≀ ◡ ( 𝑅 ⋉ 𝑆 ) = ( ≀ ◡ 𝑅 ∩ ≀ ◡ 𝑆 )
5	4	sseq1i	⊢ ( ≀ ◡ ( 𝑅 ⋉ 𝑆 ) ⊆ I ↔ ( ≀ ◡ 𝑅 ∩ ≀ ◡ 𝑆 ) ⊆ I )
6	3 5	bitri	⊢ ( Disj ( 𝑅 ⋉ 𝑆 ) ↔ ( ≀ ◡ 𝑅 ∩ ≀ ◡ 𝑆 ) ⊆ I )