Theorem rnin

Description: The range of an intersection belongs the intersection of ranges. Theorem 9 of Suppes p. 60. (Contributed by NM, 15-Sep-2004) Avoid ax-pr and ax-sep . (Revised by Umit Teoman Dogan, 10-Jun-2026)

	Ref	Expression
Assertion	rnin	⊢ ran ( 𝐴 ∩ 𝐵 ) ⊆ ( ran 𝐴 ∩ ran 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		inss1	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴
2	1	rnssi	⊢ ran ( 𝐴 ∩ 𝐵 ) ⊆ ran 𝐴
3		inss2	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐵
4	3	rnssi	⊢ ran ( 𝐴 ∩ 𝐵 ) ⊆ ran 𝐵
5	2 4	ssini	⊢ ran ( 𝐴 ∩ 𝐵 ) ⊆ ( ran 𝐴 ∩ ran 𝐵 )