Image under Subset of Relation is Subset of Image under Relation/Corollary

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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 $x \in S$.

Then:

$\map {\RR_2} x \subseteq \map {\RR_1} x$

where $\map {\RR_1} x$ denotes the image of $x$ under $\RR_1$.

Proof

\(\ds \map {\RR_2} x\)

\(=\)

\(\ds \RR_2 \sqbrk {\set x}\)

Image of Singleton under Relation

\(\ds \)

\(\subseteq\)

\(\ds \RR_1 \sqbrk {\set x}\)

Image under Subset of Relation is Subset of Image under Relation

\(\ds \)

\(=\)

\(\ds \map {\RR_1} x\)

Image of Singleton under Relation

$\blacksquare$