You missed a parenthesis- I think you mean f(a1)∩f^-1(b2)=f(a1)∩b2.

But now I have a problem with the domain and range of f. In order that the left side be non-empty there must be x that is in both f(a1) and in f^-1(b2). a1 must be a subset of the domain of f so f(x) is in the range of f. But b2 must be a subset of the range of f so that f^-1(x) is in the domain of f. That is, f must map some set onto itself.