Absoluteness of uniformization Let $E\in L$ be a $\Pi^1_1$ relation on $\mathbb R$. By the $\Pi^1_1$-uniformization theorem, there is $f\in L$ such that $f$ is a $\Pi^1_1$ function, $f\subseteq E$ and $dom(f)=dom(E)$ (that is, $L\models "dom(f)^L=dom(E)^L"$). By absoluteness, $V\models "f^V \text{ is a function and } f^V\subseteq E^V"$.


*

*Is it also true that $V\models dom(f)^V=dom(E)^V$?

*If the above is not true in general, are there any extra assumptions that will make it true?

 A: This depend what exactly you mean by the domain are equal. 
If you let $R(x,y)$ if and only if $x = y$. Then the identity function uniformizes. However the domain of this function is not the same in $L$ as it is in $V$ if $\mathbb{R}^L \neq \mathbb{R}^V$. 
However in this case, the domain has a definition. It is the set of all reals. The domain of the uniformization continues to satisfy this definition in $V$ or in $L$, although they may not literally be the same set.

However, it is possible that in $L$ the domain of a uniformization has a simple definition that does not hold of the domain in $V$:
Note that relations on $\omega$ can be coded by reals. Thus you can code countable structures in the language of set theory whose domain is $\omega$. 
Consider the relation $R(x,y)$ which states that $y$ is a wellfounded models of $\mathsf{ZF - P}$ and $\mathsf{V = L}$ and $x \in y$. 
Note that $y$ is construed as a $\dot\in$-structure on $\omega$. $y$ has a copy of $\omega$ inside. $x$ is considered as a subset of $\omega$. $x \in y$ means there is a real in $y$ using its $\omega$ that is equal to the real $x$ on the outside. 
This relation is $\Pi_1^1$ and $\mathrm{dom}(R) = \mathbb{R}^L$. In $L$, the domain is the set of all reals, but in $V$ its domain is only the set of constructible reals.  

Suppose that $f$ is $\Pi_1^1$. For all $x,y \in \mathbb{R}^L$, $f(x) = y$ is $\Pi_1^1$ so by Mostowski absoluteness, $f(x) = y$ is still true in $V$. Similarly, $E(x,f(x))$ is still true in $V$ if it was true in $L$.

Note also that from the proof of $\Pi_1^1$ uniformization, given the the $\Pi_1^1$ code for the relation, there a procedure for obtain the $\Pi_1^1$ code for a function which is provably a uniformization for the the relation. So in the sense of Noah comment, the answer is yes. (I think the procedure only depends on the $\Delta_1^1$ reduction of your relation into $\mathrm{WO}$.)
