Union of Set of Ordinals is Ordinal/Corollary

Theorem
Let $y$ be a set.

Let $\operatorname{On}$ be the class of all ordinals.

Let $F: y \to \operatorname{On}$ be a mapping.

Then:


 * $\displaystyle \bigcup F(y) \in \operatorname{On}$

where $F(y)$ is the image of $y$ under $F$.

Proof
By the Axiom of Replacement, $F(y)$ is a set.

Thus by the Axiom of Union, $\bigcup F(y)$ is a set.

By Union of Subset of Ordinals is Ordinal, $\bigcup F(y)$ is transitive and well-ordered by the epsilon relation, and is thus a member of $\operatorname{On}$, the ordinal class.