Set theory - is this a set? Suppose that $A,B$ are sets. Is
$$
\{ y \in B | \exists x \in A: y \in x \}
$$
a set? I tried to find a formal proof (replacement? comprehension?) or to show that it doesn't exist (regularity?), but I'm not really familiar with this stuff. Of course it must have to do something with ur-elements or power sets, but I'm to confused now to figure out what is going on.
In the same spirit: If $A$ is a set, is it meaningful to ask whether there exists a set $\Omega$ such that $A=\mathbb{P}(\Omega)$?
[edit]: Hm, what about:If $A$ is a set, is it meaningful to ask whether there exists a set $\Omega$ such that $A\subseteq\mathbb{P}(\Omega)$?

About the second question. It is meaningful to ask. The answer is obviously false in general. 
