Image under Subset of Relation is Subset of Image under Relation

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Theorem

Let $S$ and $T$ be sets.

Let $\RR_1 \subseteq S \times T$ be a relation in $S \times T$.

Let $\RR_2 \subseteq \RR_1$.

Let $A \subseteq S$.


Then:

$\RR_2 \sqbrk A \subseteq \RR_1 \sqbrk A$

where $\RR_1 \sqbrk A$ denotes the image of $A$ under $\RR_1$.


Proof

\(\ds y\) \(\in\) \(\ds \RR_2 \sqbrk A\)
\(\ds \leadsto \ \ \) \(\ds \exists x \in A: \, \) \(\ds \tuple {x, y}\) \(\in\) \(\ds \RR_2\) Definition of Image of Subset under Relation
\(\ds \leadsto \ \ \) \(\ds \exists x \in A: \, \) \(\ds \tuple {x, y}\) \(\in\) \(\ds \RR_1\) Definition of Subset
\(\ds \leadsto \ \ \) \(\ds y\) \(\in\) \(\ds \RR_1 \sqbrk A\) Definition of Image of Subset under Relation

$\blacksquare$


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