The Union and Product of a Family of Sets Indexed by a Set, and the Intersection of a Family with an Inhabited Index Class, Are Sets

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $I$ be a class and let $(A_{i})_{i\in I}$ be a family indexed by $I$, given by the function $A$.

¶ If $I$ is a set, then the image $A[I]$ of $I$ under $A$ is a set, and the union $\bigcup_{i\in I}A_{i}$ equals its union set $\bigcup(A[I])$; in particular $\bigcup_{i\in I}A_{i}$ is a set.

¶ If $I$ has at least one element, the intersection $\bigcap_{i\in I}A_{i}$ is a set.

¶ If $I$ is a set, the product $\prod_{i\in I}A_{i}$ is a set.