Inverse function of intersection = intersection of inverse functions

From Rosen sixth edition, page 148, section 2.3, 40.b.

Let f be a function from A to B. Let S and T be subsets of B. Show that

f inverse(S intesect T) = f inverse (S) intersecting f inverse (T)

I think I'm starting to get the hang of this but please advise on the correctness!

f:A->B S subset of B, T subset of B

Let a be an arbitrary element of f inverse ( S intersecting T) then f(a) is an element of S intersecting T

so f(a) element of S AND f(a) element of T

then a element of f inverse (S) AND a element of f inverse (T)

so a element of (f inverse (S) AND f inverse (T))

finally a element of f inverse (S) intersecting f inverse (T)

Since a is an arbitrary element of f inverse (S intersecting T) and

also of (F inverse (S) intersecting f inverse (T)) this completes the proof.

Please tell me if and where I've gone wrong.

Thanks