direct images of analytic sets are analytic

For measurable spaces $(X,ℱ)$ and $(Y,𝒢)$, consider a measurable function $f:X\to Y$. By definition, the inverse image $f^{-1}(A)$ will be in $ℱ$ whenever $A$ is in $𝒢$.However, the situation is more complicated for direct images (http://planetmath.org/DirectImage), which in general do not preserve measurability. However, as stated by the following theorem, the class of analytic subsets of Polish spaces is closed under direct images.

Theorem.

Let $f\mathrm{:}X\mathrm{\to }Y$ be a Borel measurable function between Polish spaces $X$ and $Y$. Then, the direct image $f\mathit{}\mathrm{(}A\mathrm{)}$ is analytic whenever $A$ is an analytic subset of $X$.