Image of Element is Subset

Theorem

Let $S$ and $T$ be sets.

Let $\RR \subseteq S \times T$ be a relation.

Let $A \subseteq S$.

Then:

$s \in A \implies \map \RR s \subseteq \RR \sqbrk A$

Proof

From Image of Singleton under Relation:

$\map \RR s = \RR \sqbrk {\set s}$

From Singleton of Element is Subset:

$s \in A \implies \set s \subseteq A$

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

$\blacksquare$