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	$$⊢\mathrm{ran}\left(A\cap B\right)\subseteq \mathrm{ran}A\cap \mathrm{ran}B$$

Proof

Step	Hyp	Ref	Expression
1		inss1	$$⊢A\cap B\subseteq A$$
2	1	rnssi	$$⊢\mathrm{ran}\left(A\cap B\right)\subseteq \mathrm{ran}A$$
3		inss2	$$⊢A\cap B\subseteq B$$
4	3	rnssi	$$⊢\mathrm{ran}\left(A\cap B\right)\subseteq \mathrm{ran}B$$
5	2 4	ssini	$$⊢\mathrm{ran}\left(A\cap B\right)\subseteq \mathrm{ran}A\cap \mathrm{ran}B$$