Let $V$ be a transitive model of ZFC, $\mathbb{P} \in V$ be a forcing poset such that $(|\mathbb{P}| < \delta)^{V}$ for some regular cardinal $\delta$ in $V$, and $G$ be a $V$-generic filter over $\mathbb{P}$. I wish to show that $\langle V, V[G]\rangle$ has the $\delta$-approximation property. That is, for every $X \in V[G]$ such that $X \subset V$, if $Y \cap X \in V$ for every $Y \in V$ with $(|Y| < \delta)^V$, then $X \in V$.
In "Certain very large cardinals are not created in small forcing extensions" (page 3), Laver gives the following argument.
Let $A$ be a set of ordinals in $V[G] \setminus V$. In $V$, we define $B$ as follows.
$$B = \{\alpha : \text{for some } r\in \mathbb{P}, \alpha \text{ is the least ordinal such that for some forcing conditions } p , q > r, p \Vdash \alpha \in A \land q \Vdash \alpha \notin A\}$$
$B$ must be nonempty (otherwise $A$ is definable in $V$) and have size $<\delta$. Then Laver claims that $B\cap A \notin V$. My first question is that I don't understand why $B\cap A \notin V$. It'd appreciated if anyone could explain it in detail.
My second question is: why does it suffice to show that $\delta$-approximation holds for sets of ordinals in $V[G]$? I attempt to show this by using the fact that every set is coded by a set of ordinals. So let $X \in V[G] \setminus V$ with $X\subset V$ be an arbitrary set coded by a set of ordinals $A \in V[G]\setminus V$. Then by the argument above, there must be some small set $B \in V$ such that $B \cap A \notin V$. But how do we know $B \cap A$ codes a small set of $X$?
For the first question, note that for any $r\in G$, there is $\alpha\in B$ that "comes from $r$". For any such $\alpha$, $G$ has "generically decided" whether to put $\alpha$ into $A$ or not and this makes sure that $B\cap A\notin V$. For suppose $C\in V$ is a set of ordinals. Then for any $r\in G$ there is $\alpha\in B$ and some $p\leq r$ so that $$p\Vdash\check\alpha\in A\Leftrightarrow\alpha\notin C$$ This is a density argument that shows that there must be some $r', p'\in G$ and some $\alpha'\in B$ that "comes from $r'$" so that $$p'\Vdash\alpha'\in A\Leftrightarrow \alpha'\notin C$$ and hence $B\cap A\neq C$.
For the second question, use the following trick: Since $X\subseteq V$, there is some $Y\in V$ with $X\subseteq Y$. Any $V_\alpha$ with $\alpha$ large enough works. Now in $V$ (!) code $Y$ as a set of ordinals, actually any bijection $$\pi:Y\rightarrow \gamma$$ for an ordinal $\gamma$ is fine. Now replace $X$ by $X'=\pi[X]$.