Image of Element is Subset

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Theorem

Let $S$ and $T$ be sets.

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

Let $A \subseteq S$.


Then:

$s \in A \implies \mathcal R \left({s}\right) \subseteq \mathcal R \left[{A}\right]$


Proof

From Image of Singleton under Relation:

$\mathcal R \left({s}\right) = \mathcal R \left[{\left\{{s}\right\}}\right]$

From Singleton of Element is Subset:

$s \in A \implies \left\{{s}\right\} \subseteq A$

The result follows from Image of Subset is Subset of Image.

$\blacksquare$