Axiom of Replacement for Classes

Let $Y$ be a class such that, for all sets $v$, $u$ and $u'$, if the ordered pairs satisfy $(v,u)\in Y$ and $(v,u')\in Y$, then $u=u'$. Then for every set $x$ there is a set $y$ such that, for every set $u$, $u\in y$ if and only if there is a set $v$ with $v\in x$ and $(v,u)\in Y$.