Combinatorial Properties and Dependent choice in symmetric extensions based on L{e}vy Collapse

We work with symmetric extensions based on L{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $langle mathbb{P},mathcal{G},mathcal{F}rangle$, if $mathbb{P}$ is $kappa$-distributive and $mathcal{F}$ is $kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{kappa}$, then $DC_{<kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $langle mathbb{P},mathcal{G},mathcal{F}rangle$, if $mathbb{P}$ is $kappa$-strategically closed and $mathcal{F}$ is $kappa$-complete.