Studying descriptive set theory, a found the Mostowski absoluteness theorem: if $\phi$ is $\Delta_{1}$, then $\forall x\in \omega^{\omega} \phi(x)$ is absolute between $V$ and $L$.
I notice that $\forall x\in \kappa^{\gamma}\exists y\in \gamma\, \phi(y,x(y))$ is also absolute between $V$ and $L$ for any ordinals $\kappa$ and $\gamma$ and $\kappa^{\gamma}:=\lbrace f|f:\gamma \rightarrow \kappa \rbrace$. I wanna know if sentences $\forall x\in \kappa^{\gamma}\, \phi(x)$ are also absolute.
No, a statement of the form $\forall x\in\kappa^\gamma\phi(x)$ is not absolute in general.
To see this, let $\gamma=\omega_1^L$, $\kappa=\omega$ and let $\phi(x)$ be the sentence "$x$ is not one-to-one." Then $L$ thinks $\forall x\in\omega^{\omega_1^L} \phi(x)$ holds. Furthermore, $\phi$ is $\Delta_0$.
Now let $L[G]$ be a generic extension of $L$ obtained by collapsing $\omega_1^L$ to a countable ordinal. Then $L[G]$ thinks there is an one-to-one map $f\colon \omega_1^L\to \omega$.