Parametric bing and Krasinkiewicz maps

It is shown that if is a perfect map between metrizable spaces and Y is a C-space, then the function space C(X,I) with the source limitation topology contains a dense Gδ-subset of maps g such that every restriction map gy=g|f−1(y), y∈Y, satisfies the following condition: all fibers of gy are hereditarily indecomposable and any continuum in f−1(y) either contains a component of a fiber of gy or is contained in a fiber of gy.