I don't see a case involved in the proof of Proposition 10.20 in Kanamori's book.
Let $P$ be a separative poset with $|P|\leq|\alpha|$, for an ordinal $\alpha$. Suppose $P$ forces that there is a surjective function $f:\omega\rightarrow \alpha$ with $f\notin V$. The claim is that for any $p\in P$ there is a maximal antichain below $p$ of cardinality $|\alpha|$. We have two cases:
- $|\alpha|>\omega$,
- $\alpha$ is countable.
Case 1 is clear: in the extension $\alpha$ is countable since $f$ is surjective. So $P$ is not $|\alpha|$-c.c., proving the existence of a maximal antichain $\leq p$ of size $|\alpha|$.
For case 2, I don't see the implication stated in the book: if $|\alpha|=\omega$, this follows from $\Vdash f\notin\check{V}$ since any condition must have incompatible extensions. Where do we use the fact that $\Vdash f\notin\check{V}$?