My question is about a proof of **Theorem 4.12** given in the text *Introduction to Set Theory* by Hrbacek and Jech.

The authors first prove **Theorems 4.9**, **4.10**, and **4.11**. For reference, I quote these theorems below.

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> 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.
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**Theorem 4.9: Transfinite Recursion Theorem**

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> There exists a class function $F:\operatorname{Ord}\to V$ such that $F(\alpha)=G(F\restriction \alpha)$ for all $\alpha\in\operatorname{Ord}$.
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**Theorem 4.10: Transfinite Recursion Theorem, Parametric Version**

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> There exists a class function $F:\operatorname{Ord}\to V$ such that $F(z,\alpha)=G(z,F\_z\restriction \alpha)$ for all $\alpha\in\operatorname{Ord}$ and $z\in V$ where $F\_z\restriction \alpha:=\{\langle\beta,F(z,\beta)\rangle\mid\beta<\alpha\}$.
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**Theorem 4.11:**

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> Let $G\_1,G\_2,G\_3$ be class functions from $V$ to $V$. There exists a class function $F:\operatorname{Ord}\to V$ such that
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> (1) $F(0)=G\_1(\emptyset)$
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> (2) $F(\alpha+1)=G\_2(F(\alpha))$ for all $\alpha\in\operatorname{Ord}$
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> (3) $F(\alpha)=G\_3(F\restriction\alpha)$ for all limit $\alpha\neq 0$
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Then they say *A parametric version of **Theorem 4.11** is straightforward and we leave it to the reader*. From my attempt, the **parametric version of Theorem 4.11** is as follows (Honestly, I'm not sure if my attempt is correct or not)

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> Let $G\_1,G\_2,G\_3$ be class functions from $V$ to $V$. There exists a class function $F$ such that, for all $z\in V$
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> $F(z,0)=G\_1(z,\emptyset)$
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> $F(z,\alpha+1)=G\_2(z,F\_z(\alpha))$ for all $\alpha\in\operatorname{Ord}$
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> $F(z,\alpha)=G\_3(z,F\_z\restriction\alpha)$ for all limit $\alpha\neq 0$
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Next they make use of this **parametric version of Theorem 4.11** in the proof of **Theorem 4.12**.

**Theorem 4.12:**

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> For any set $a$, there is a unique infinite sequence $(f\_n\mid n\in \Bbb N)$ such that
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> (1) $f\_0=a$
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> (2) $f\_{n+1}=G(f\_n,n)$ for all $n\in \Bbb N$
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The proof for **Theorem 4.12** given in the text is as follows:

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> Let $G:V\to V$ be a class function. We want to find, for every set $a$, a sequence $(f\_n\mid n\in \Bbb N)$ such that $f\_0=a$ and $f\_{n+1}=G(f\_n,n)$ for all $n\in \Bbb N$. By the **parametric version of the Transfinite Recursion Theorem 4.11**, there is class function $F$ such that $F(0)=a$ and $F(n+1)=G(F(n),n)$ for all $n\in \Bbb N$. Now we apply the Axiom of Replacement: There exists a sequence $(f\_n\mid n\in \Bbb N)$ that is equal to $F\restriction \omega$ and the Theorem follows.
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I don't understand how the **parametric version of Theorem 4.11** is used to derive the class function $F$ in the proof of **Theorem 4.12**. Can someone please help me fill in the blanks?