Union of Set of Ordinals is Ordinal/Corollary/Proof 1

Proof
By the Axiom of Replacement, $\map F y$ is a set.

Thus by the Axiom of Unions, $\bigcup \map F y$ is a set.

By Union of Set of Ordinals is Ordinal, $\bigcup \map F y$ is transitive.

By the epsilon relation $\bigcup \map F y$ is well-ordered.

Thus $\bigcup \map F y$ is a member of $\On$, the class of all ordinals.