For a σ-ideal I of sets in a Polish space X and for A ⊆ $X^2$, we consider the generalized projection (A) of A given by (A) = x ∈ X: Ax ∉ I, where $A_x$ =y ∈ X: 〈x,y〉∈ A. We study the behaviour of with respect to Borel and analytic sets in the case when I is a ${\sum }_2^0$-supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that [${\sum }_1^1\left(X^2\right)]={\sum }_1^1\left(X\right)$ for a wide class of ${\sum }_2^0$-supported σ-ideals.