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<κ can be preserved in the symmetric extension of V in terms of
symmetric system ⟨P,G,F⟩, if
P is κ-distributive and F is κ-complete.
Further we observe that if V is a model of ZF + DCκ, then
DC<κ can be preserved in the symmetric extension of V in terms of
symmetric system ⟨P,G,F⟩, if
P is κ-strategically closed and F is
κ-complete.Comment: Revised versio