Union of Set of Ordinals is Ordinal/Corollary/Proof 2

Proof
So:
 * $\ds \bigcup_{z \mathop \in y} \map F z \subseteq \On$

So $\ds \bigcup_{z \mathop \in y} \map F z$ is a transitive set.

Therefore $\ds \bigcup_{z \mathop \in y} \map F z$ is an ordinal.


 * $\ds \bigcup_{z \mathop \in y} \map F z = \bigcup \Img y$

Let $U$ denote the universal class.

Therefore $\ds \bigcup_{z \mathop \in y} \map F z$ is a set, so it is a member of the class of all ordinals.