Theorem relssinxpdmrn

Description: Subset of restriction, special case. (Contributed by Peter Mazsa, 10-Apr-2023)

	Ref	Expression
Assertion	relssinxpdmrn	⊢ ( Rel 𝑅 → ( 𝑅 ⊆ ( 𝑆 ∩ ( dom 𝑅 × ran 𝑅 ) ) ↔ 𝑅 ⊆ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		relssdmrn	⊢ ( Rel 𝑅 → 𝑅 ⊆ ( dom 𝑅 × ran 𝑅 ) )
2	1	biantrud	⊢ ( Rel 𝑅 → ( 𝑅 ⊆ 𝑆 ↔ ( 𝑅 ⊆ 𝑆 ∧ 𝑅 ⊆ ( dom 𝑅 × ran 𝑅 ) ) ) )
3		ssin	⊢ ( ( 𝑅 ⊆ 𝑆 ∧ 𝑅 ⊆ ( dom 𝑅 × ran 𝑅 ) ) ↔ 𝑅 ⊆ ( 𝑆 ∩ ( dom 𝑅 × ran 𝑅 ) ) )
4	2 3	bitr2di	⊢ ( Rel 𝑅 → ( 𝑅 ⊆ ( 𝑆 ∩ ( dom 𝑅 × ran 𝑅 ) ) ↔ 𝑅 ⊆ 𝑆 ) )