Prove: If $\displaystyle f: A \rightarrow B $ and $\displaystyle D_1 \subset D_2 $ are subsets of B, then $\displaystyle f^{-1} (D_1) \subset f^{-1} (D_2)$.
It suffices to show that any $\displaystyle y\in f^{-1}(D_1)$ happens to be an element of $\displaystyle f^{-1}(D_2)$ as well:
So let's assume that $\displaystyle y\in f^{-1}(D_1)$. This means that $\displaystyle f(y)\in D_1$. Because of $\displaystyle D_1\subseteq D_2$, it follows that $\displaystyle f(y)\in D_2$. But this is equivalent to saying that $\displaystyle y\in f^{-1}(D_2)$.