Image of Set Difference under Relation/Corollary 1

Corollary to Image of Set Difference under Relation
Let $\mathcal R \subseteq S \times T$ be a relation.

Let $A \subseteq B \subseteq S$.

Then:
 * $\complement_{\mathcal R \left[{B}\right]} \left({\mathcal R \left[{A}\right]}\right) \subseteq \mathcal R \left[{\complement_B \left({A}\right)}\right]$

where:
 * $\mathcal R \left[{B}\right]$ denotes the image of $B$ under $\mathcal R$
 * $\complement$ (in this context) denotes relative complement.

Proof
We have that $A \subseteq B$.

Then by definition of relative complement:
 * $\complement_B \left({A}\right) = B \setminus A$
 * $\complement_{\mathcal R \left[{B}\right]} \left({\mathcal R \left[{A}\right]}\right) = \mathcal R \left[{B}\right] \setminus \mathcal R \left[{A}\right]$

Hence, when $A \subseteq B$:
 * $\complement_{\mathcal R \left[{B}\right]} \left({\mathcal R \left[{A}\right]}\right) \subseteq \mathcal R \left[{\complement_B \left({A}\right)}\right]$

means exactly the same thing as:
 * $\mathcal R \left[{B}\right] \setminus \mathcal R \left[{A}\right] \subseteq \mathcal R \left[{B \setminus A}\right]$