Given classes $H$, $R$, can we find a class $G$ such that the following statement is provable in ZFC?

If $R\subseteq V\times V$ and $H$ is a function from $V$ into $Ord$ such that for all sets $x,y$, $y\ R\ x$ implies that $H(y)<H(x)$, then $G$ is a function from $V$ into $V$ such that for all sets $x$, $G(x)=\{G(y)\ |\ y\ R\ x\}$.

Note that the relation $R$ in the above statement must be well-founded, but need not to be set-like.