All models of $\mathsf{ZFC}$ between $V$ and $V[G]$ are generic extensions of $V$

Question

I'm reading the proof of lemma 15.43 of Jech's Set Theory:

Let $G$ be generic on a complete Boolean algebra $B$. If $M$ is a model of $\mathsf{ZFC}$ such that $V\subset M\subset V[G]$, then there exists a complete subalgebra $D\subset B$ such that $M=V[D\cap G]$.

In the proof of this theorem the author states the following fact: "First we note that since $M$ satisfies the Axiom of Choice, there is for every $X\in M$ a set of ordinals $A_X\in M$ such that $X\in V[A_X].$"

My question is, why is this fact true?

Thanks

Answer 1

Using the axiom of choice $(TC(\{X\}),\in)$ can be encoded as $(\alpha,R)$ where $\alpha$ is an ordinal and $R\subseteq\alpha\times\alpha$. Now using encoding of pairs of ordinals from $V$ (or even $L$) we can replace $R$ by some $A\subseteq\alpha$ altogether.

Now consider in $V[A]$. We can decode $R$ back from $A$ because the encoding function was in $V$, so $(\alpha,R)\in V[A]$. Now consider the transitive collapse of this structure, it has to be $(TC(\{X\}),\in)$. So $X\in V[A]$ as well.