These are statement 4 of example 2.3.6 and its solution from section 2.3 in Daniel J. Velleman's "How to Prove It - A Structured Approach" (great book), where the author asks the reader to analyze the logical form of several statements. On the solution to this particular statement, i.e.:
$x\in\cup\{\mathcal{P}(A)|A\in\mathcal{F}\}$
he argues that, according to the definition of union given earlier:
$\cup\mathcal{F}=\{x|\exists A\in\mathcal{F}(x\in A)\}=\{x|\exists A(A\in\mathcal{F}\wedge x\in A)\}$
the statement means that "... $x$ is an element of at least one of the sets $\mathcal{P}(A)$, for $A\in\mathcal{F}$. In other words, $\exists A\in\mathcal{F}(x\in\mathcal{P}(A))$."
Intuitively it makes sense, but I can't write it down formally. If I state that $x\in\cup\mathcal{F}$ is true, I know that $\exists A\in\mathcal{F}(x\in A)$. But I get lost when I try to replace $\mathcal{F}$ with $\mathcal{P}(A)$, and can't figure out the rest.