This may be a very trivial question, but I cannot overcome my doubt. I am trying to prove that $V_\alpha=\bigcup_{\beta<\alpha}\wp(V_\beta)$ This can be done by transfinite induction. What I miss is the following step
$K=\{\wp(V_\beta) | \beta<\alpha\}$ is a set
Now I understand that by inductive hypothesis plus the power set axiom each element of the collection is a set. Moreover, the collection is indexed by a set, the ordinal $\alpha$. But I cannot find a set to include this collection in. I would try with something like $K\subset\wp(\alpha\times ?)$ but cannot replace the ?.
Thanks in advance
Consider the following predicate of two variables: $$\mathcal Q(x,y):=[ (x\in\alpha\land y=\wp(V_x))\lor (x\notin \alpha\land y=\emptyset)]$$
Now, for all sets $x,y,z$, $$\mathcal Q(x,y)\land \mathcal Q(x,z)\implies y=z$$
and therefore, by the axiom schema of replacement, there is some set $U$ containing all the sets $y$ such that there is some $x\in\alpha$ satisfying $\mathcal Q(x,y)$.
$K$ is of course a subset of any such set.