# Image under Subset of Relation is Subset of Image under Relation

## Theorem

Let $S$ and $T$ be sets.

Let $\mathcal R_1 \subseteq S \times T$ be a relation in $S \times T$.

Let $\mathcal R_2 \subseteq \mathcal R_1$.

Let $A \subseteq S$.

Then:

$\mathcal R_2 \left[{A}\right] \subseteq \mathcal R_1 \left[{A}\right]$

where $\mathcal R_1 \left[{A}\right]$ denotes the image of $A$ under $\mathcal R_1$.

## Proof

 $\displaystyle y$ $\in$ $\displaystyle \mathcal R_2 \left[{A}\right]$ $\quad$ $\quad$ $\displaystyle \leadsto \ \$ $\, \displaystyle \exists x \in A: \,$ $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \mathcal R_2$ $\quad$ Definition of Image of Subset under Relation $\quad$ $\displaystyle \leadsto \ \$ $\, \displaystyle \exists x \in A: \,$ $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \mathcal R_1$ $\quad$ Definition of Subset $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle y$ $\in$ $\displaystyle \mathcal R_1 \left[{A}\right]$ $\quad$ Definition of Image of Subset under Relation $\quad$

$\blacksquare$