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)

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

Proof

Step	Hyp	Ref	Expression
1		cnvin	⊢ ◡ ( 𝐴 ∩ 𝐵 ) = ( ◡ 𝐴 ∩ ◡ 𝐵 )
2	1	dmeqi	⊢ dom ◡ ( 𝐴 ∩ 𝐵 ) = dom ( ◡ 𝐴 ∩ ◡ 𝐵 )
3		dmin	⊢ dom ( ◡ 𝐴 ∩ ◡ 𝐵 ) ⊆ ( dom ◡ 𝐴 ∩ dom ◡ 𝐵 )
4	2 3	eqsstri	⊢ dom ◡ ( 𝐴 ∩ 𝐵 ) ⊆ ( dom ◡ 𝐴 ∩ dom ◡ 𝐵 )
5		df-rn	⊢ ran ( 𝐴 ∩ 𝐵 ) = dom ◡ ( 𝐴 ∩ 𝐵 )
6		df-rn	⊢ ran 𝐴 = dom ◡ 𝐴
7		df-rn	⊢ ran 𝐵 = dom ◡ 𝐵
8	6 7	ineq12i	⊢ ( ran 𝐴 ∩ ran 𝐵 ) = ( dom ◡ 𝐴 ∩ dom ◡ 𝐵 )
9	4 5 8	3sstr4i	⊢ ran ( 𝐴 ∩ 𝐵 ) ⊆ ( ran 𝐴 ∩ ran 𝐵 )