Image of Union under Mapping/Family of Sets

Theorem
Let $S$ and $T$ be sets.

Let $\left\langle{S_i}\right\rangle_{i \in I}$ be a family of subsets of $S$.

Let $f: S \to T$ be a mapping.

Then:
 * $\displaystyle f \left({\bigcup_{i \mathop \in I} S_i}\right) = \bigcup_{i \mathop \in I} f \left({S_i}\right)$

where $\displaystyle \bigcup_{i \mathop \in I} S_i$ denotes the union of $\left\langle{S_i}\right\rangle_{i \in I}$.

Proof
As $f$, being a mapping, is also a relation, we can apply Image of Union/Family of Sets:


 * $\displaystyle \mathcal R \left({\bigcup_{i \mathop \in I} S_i}\right) = \bigcup_{i \mathop \in I} \mathcal R \left({S_i}\right)$