I've heard someone saying that in $L(\mathbb{R})$ the following form of $\text{CH}$ holds:
$L(\mathbb{R})\vDash \forall X\subseteq \mathbb{R}(X \text{ countable or } \mathbb{R}\le^* X)$, i.e. every uncountable subset of the reals surjects onto $\mathbb{R}$
Now I'm wondering whether this is a theorem of $\text{ZF(C)}$ or we need to assume some large cardinal to prove it (e.g. so to have $\text{AD}$).
I'm struggling to find some reference.
Any help?
Thanks!
Edit: It cannot be a theorem of $\text{ZF}$ as otherwise we could prove that $\text{ZF}+\text{DC}+\text{PSP}$ is equiconsistent with $\text{ZF}$, which is not. But I'm still not finding a reference for the result under $\text{AD}$ for example..