Borel sets with countable sections for nonseparable spaces

We prove that every (extended) Borel subset E of X x Y, where X is a complete metric and Y is Polish, can be covered by countably many extended Borel graphs of mappings from X to Y if the sections E-x = {y is an element of Y : ( x, y). E}, x is an element of X, are countable. This is a nonseparable version of a classical theorem of Luzin and Novikov.