I don't know if this is explicitly written down anywhere, maybe only mentioned in passing or implicit in the usual exposition of the construction of the Solovay model. Here's one way to do it:
First one proves an alternative characterization of the perfect set property for co-analytic sets (in symbols, $\mathsf{PSP}(\mathbf{\Pi^1_1})$):
Theorem. The following are equivalent:
- $\mathsf{PSP}(\mathbf{\Pi^1_1})$
- $\mathsf{PSP}(\mathbf{\Sigma^1_2})$
- for every real $x$, the set of real numbers in $L[x]$ is countable.
This characterization can be found in many standard texts (it's Theorem 25.38 in Jech, for instance). It involves careful descriptive set theoretic analysis of how these sets of reals are represented by special kinds of trees.
Having this in hand, one can start with an inaccessible cardinal $\kappa$ and use Levy collapse to render every ordinal below $\kappa$ countable, and $\kappa$ becomes $\omega_1$ in the extension. Working in the extension, one can show that 3 holds.
This is proven by appealing to the following property of the Levy collapse: if a real number is added by the forcing, then it is already added in an intermediate stage when $\kappa$ is still inaccessible (see Kanamori 10.21 for example). So whatever the cardinality of $\mathbb{R}\cap L[x]$ is by that stage, it is $<\kappa$ and will eventually be rendered countable by the forcing.
Note that the starting assumption of an inaccessible cardinal cannot be dispensed with, because when 3 holds, $(\omega_1)^V$ will be an inaccessible cardinal in $L$, for instance.
