Closed subsets of the domain whose image has the dimension of the range

It is shown that if dim Y < ∞ and if f(X) = Y is a mapping between compact metric spaces such that 1 ? m ? dim f^-1(y)?n for all y ? Y, then there exists a closed set K ? X such that dim K ? n ? m and dim f(K) = dim Y. This answers a question posed by J. Keesling and D. Wilson.