Can the following uniformization statement be proved by $ZF+AD+DC$?



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> For any binary relation $R\subseteq \mathbb{R}^2$ with the property that $\forall x (\{y\mid R(x,y)\}\mbox{ is at most countable and nonempty})$, then there is a function $f:\mathbb{R}\to \mathbb{R}$ uniformizing $R$.
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Uniformization under AD
Here's an argument under the further assumption that $V=L(\mathbb{R})$. Something like this is presumably recorded somewhere. The main point comes from Steel's paper "Scales in $L(\mathbb{R})$", but I don't find it quite explicitly there.


Since $V=L(\mathbb{R})$, we can fix a real $x\_0$ such that our relation $R$ is $\mathrm{OD}\_{x\_0}$. Then for each $x\in\mathbb{R}$,
$R\_x=\{y\bigm|R(x,y)\}$ is $\mathrm{OD}\_{(x\_0,x)}$. It suffices
to see that $R\_x$ contains some element which is $\mathrm{OD}\_{(x\_0,x)}$,
since then we can set $f(x)=$ the least such (in the standard ordering of $\mathrm{OD}\_{(x\_0,x)}$).


In fact, for each real $x$, every countable $\mathrm{OD}\_x$ set of reals is $\subseteq\mathrm{OD}\_x$. For let's assume $x=\emptyset$; the relativization to other $x$ is routine. Fix a $\Sigma\_1$ formula $\varphi$ such that for some ordinals $\alpha<\beta$, the set $$A\_{\alpha\beta}=\{y\in\mathbb{R}\bigm|L\_\beta(\mathbb{R})\models\varphi(\alpha,\mathbb{R},y)\}$$ is countable; each OD set is of this form. We want to see that each such $A\_{\alpha\beta}\subseteq\mathrm{OD}$ (assuming it's countable). Let $A'\_{\alpha\beta}=A\_{\alpha\beta}$ if $A\_{\alpha\beta}$ is countable, and $A'\_{\alpha\beta}=\emptyset$ otherwise.


Claim: $A=\bigcup\_{\alpha\beta}A'\_{\alpha\beta}$ is countable.


Proof: Otherwise we can define a prewellorder $<^\*$ on $A$ such that the set of $<^\*$-predecessors of any given real is countable, and then argue like in the proof that there is no $\omega\_1$-sequence of pairwise distinct reals. QED (Claim).


Note now that $A$ is $\Sigma\_1^{L(\mathbb{R})}$ (here the standard notation allows the parameter $\mathbb{R}$ as default, but this is otherwise lightface, i.e. no other parameters). But by Proposition 2.11 of Steel's "Scales in $L(\mathbb{R})$" (applied at some sufficiently large and reflective level $\alpha\in\mathrm{OR}$), and since $A$ is countable, it follows that $A\subseteq\mathrm{OD}$, as desired.


