Let $f: A \rightarrow B$ be a map of sets, and let {$X_i$}$_{i\in I}$ be an indexed collection of subsets of A. Prove that $f(\cap_{i\in I}X_i)\subset\cap_{i\in I}f(X_i)$
From a second year calculus course. Have only done Basic Set Theory and Injective/Surjective functions. I'm thinking the proof has something to do with the injectivity of $f(\cap_{i\in I}X_i)$? 
Any help would be appreciated, thanks.

Use the fact that 
$$
X\subset Y\implies f(X)\subset f(Y)
$$
Also for any $i\in I$, there is
$$
\bigcap_{i\in I}X_i\subset X_i
$$
Thus for any $i\in I$, we have
$$
f\left(\bigcap_{i\in I}X_i\right)\subset f(X_i)\tag1
$$
Since $(1)$ holds for any $i\in I$
$$
f\left(\bigcap_{i\in I}X_i\right)\subset \bigcap_{i\in I}f(X_i)
$$
