Image of Subset under Relation equals Union of Images of Elements

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

Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$.

Let $X \subseteq S$ be a subset of $S$.

Then:
 * $\displaystyle \mathcal R \left[{X}\right] = \bigcup_{x \mathop \in X} \mathcal R \left({x}\right)$

where:
 * $\mathcal R \left[{X}\right]$ is the image of the subset $X$ under $\mathcal R$
 * $\mathcal R \left({x}\right)$ is the image of the element $x$ under $\mathcal R$.

Proof
By definition:
 * $\mathcal R \left[{X}\right] = \left\{ {y \in T: \exists x \in X: \left({x, y}\right) \in \mathcal R}\right\}$
 * $\mathcal R \left({x}\right) = \left\{ {y \in T: \left({x, y}\right) \in \mathcal R}\right\}$

First:

Then:

So:
 * $\displaystyle \bigcup_{x \mathop \in X} \mathcal R \left({x}\right) \subseteq \mathcal R \left[{X}\right]$

and:
 * $\displaystyle \mathcal R \left[{X}\right] \subseteq \bigcup_{x \mathop \in X} \mathcal R \left({x}\right)$

The result follows by definition of set equality.