A bootstrap from the traditional version of Transfinite Recursion to the parametric one

2018-10-23 14:51:54

I'm self-learning Transfinite Recursion Theorem and its variants from textbook Introduction to Set Theory by Hrbacek and Jech.

After the authors have presented proofs of theorems 4.9, 4.10, and 4.11, they conclude A parametric version of Theorem 4.11 is straightforward and we leave it to the reader.

I have tried to give it a shot, but I'm not sure if it's fine or not. Does my attempt look fine or contain flaws? I'm very pleased to receive suggestion!

Let $V$ be the class of all sets, $\operatorname{Ord}$ be the class of all ordinals, and $G:V\to V$ be a class function.

Theorem 4.9: Transfinite Recursion Theorem

There exists a class function $F$ such that $F(\alpha)=G(F\restriction \alpha)$ for all $\alpha\in\operatorname{Ord}$.

Theorem 4.10: Transfinite Recursion Theorem, Parametric Version

There exists a class function $F$ such that $F(z,\alpha)=G(z,F_z\restriction \alpha)$ for all $\alpha\in\operatorname{Ord}$ and $z\in V$ where $F_z\restriction \alp