Image of Union under Mapping

Theorem
Let $$f: S \to T$$ be a mapping. Let $$A$$ and $$B$$ be subsets of $$S$$. Then:

$$f \left({A \cup B}\right) = f \left({A}\right) \cup f \left({B}\right)$$

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

$$\mathcal{R} \left({A \cup B}\right) = \mathcal{R} \left({A}\right) \cup \mathcal{R} \left({B}\right)$$