In Section 4.2 of "Discovering Modern Set Theory I." by Just and Weese, The proof of Theorem 25 (Principle of Generalized Recursive Definitions) considers the set of all $z \in Z$ such that there exists a function $F_z : I(z) \to X$ such that $F(y) = G(F|I(y),y)$ for each $y \in I(z)$. I am confused by this, and not just for lack of subscripts on $F$.
1) Since a wellfounded relation need not be transitive, $F_z(y) = G(F_z|I(y),y)$ does not seem to work, because $F_z|I(y)$ may not be a total function from $I(y)$ to $X$ since $y \in I(z) \kern.6em\not\kern -.6em \implies I(y) \subseteq I(z)$.
2) $F_z(y) = G(F_y|I(y),y)$ seems more plausible, but other $F_y$ have not been introduced in the preceding prose (whereas $F_z$ is explicitly quantified).
Should $F_z$'s domain be expanded to include the "downward closure" of $z$ with respect to the well-founded relation $W$, or am I missing something?