Question concerning forms for defining sets

Hi:

I am learning set theory and would like to request some help concerning
defining sets. I have seen the forms:

A = { x : Formula(x) }
B = { x in Y : Formula(x) }

but I have not seen something like

C = { x subset Y : Formula(x) }

Is this a valid form/definition of a set? Is is a valid form if the
corresponding set D exists?

D = { x in V : Formula(x) } such that Y subset V

Thanks.

Scott

.