Image under Subset of Relation is Subset of Image under Relation

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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 $\quad$

$\blacksquare$


Sources