Let $\Gamma$ be a collection of relations on the reals and let $M$ be a set of reals. We call $M$ a perfect set basis for $\Gamma$ if every set in $\Gamma$ with parameters from $M$ which is not totally included in $M$ contains a perfect subset with code in $M$. A simple elementary proof is given of the following result (assuming mild regularity conditions on $\Gamma$ and $M$): If $M$ is a perfect set basis for $\Gamma$, the field of every wellordering in $\Gamma$ is contained in $M$. An immediate corollary is Mansfield's Theorem that the existence of a $\Sigma^1_2$ wellordering of the reals implies that every real is constructible. Other applications and extensions of the main result are also given.