Set Union can be Derived using Axiom of Abstraction

Theorem
Let $a$ be a set of sets.

By application of the, the union $\bigcup a$ can be formed.

Hence the union $\bigcup a$ can be derived as a valid object in Frege set theory.

Proof
Let $P$ be the property defined as:


 * $\forall x: \map P x := \paren {\exists y: y \in a \land x \in y}$

where $\land$ is the conjunction operator.

That is, $\map P x$ :
 * $x$ is an element of some set $y$, where $y$ is one of the sets that comprise the elements of $a$.

Hence, using the, we form the set:
 * $\bigcup a := \set {x: \exists y: y \in a \land x \in y}$