I am looking for a reference for the following result:

Theorem Assume $\kappa$ is a Woodin cardinal. Then after forcing with the Levy collapse $\mathbb{P}=Col(\omega, < \kappa)$, the $\Sigma^1_2$-determinacy holds in $V[G_{\mathbb{P}}]$.

Note that it suffices to show that $\Sigma^1_2$-determinacy holds in $L[\mathbb{R}]^{V[G_{\mathbb{P}}]}.$

I am looking for a reference for the following result:

Theorem Assume $\kappa$ is a Woodin cardinal. Then after forcing with the Levy collapse $\mathbb{P}=Col(\omega, \kappa)$, the $\Sigma^1_2$-determinacy holds in $V[G_{\mathbb{P}}]$.

Note that it suffices to show that $\Sigma^1_2$-determinacy holds in $L(\mathbb{R})^{V[G_{\mathbb{P}}]}.$