Let $\mathcal{M}$ be a countable transitive standard-model of ZFC.

Let $B \in \mathcal{M}$ be a boolean algebra that is complete in $\mathcal{M}$.

Further, let $\mathcal{M}^{(B)}$ be the corresponding boolean model of ZFC.

Now we consider an $\mathcal{M}$-generic ultrafilter $U$ on $B$.

According to Jech (Set Theory, p.216), the elements of the generic extension $\mathcal{M}[U]$ can be obtained via the map $\cdot^U: x \mapsto x^U := \lbrace{y^U \mid x(y) \in U \rbrace}$ where $x \in \mathcal{M}^{(B)}$. But when reading Bell (Set Theory Boolean-Valued models and Independence Proofs, p.91) the following map is given: $i(x) = \lbrace i(y) \mid \Vert y \in x\Vert \in U \rbrace$ where $x \in \mathcal{M}^{(B)}$. I assume that the following holds:
$$\forall x \in \mathcal{M}^{(B)}: x^U = i(x). \ \ \ [*]$$

But at the same time I am not yet fully convinced as $u(x) \leq \Vert x \in u \Vert$ for $x \in \text{dom}(u)$.

Can anyone verify that $[*]$ (or a variant of that statement) holds (and give a short reason why)?