We relate some subsets $G$ of the product $X\times Y$ of nonseparable Luzin (e.g., completely metrizable) spaces to subsets $H$ of ${\mathbb{N}}^{\mathbb{N}}\times Y$ in a way which allows to deduce descriptive properties of $G$ from corresponding theorems on $H$. As consequences we prove a nonseparable version of Kondô’s uniformization theorem and results on sets of points $y$ in $Y$ with particular properties of fibres $f^{-1}\left(y\right)$ of a mapping $fX\to Y$. Using these, we get descriptions of bimeasurable mappings between nonseparable Luzin spaces in terms of fibres.