Prove that $A \subset B$ if and only if $A \cap B = A$

Prove that $A \subset B$ if and only if $A \cap B = A$

I'm having problems writing this proof using formal logic operators. I know the idea behind it: Since $A \subset B$, the only elements A and B have in common are those of A, so the intersection must be just A...

$\Rightarrow:$ We always have $A\cap B\subseteq A$. For the reversed containment, if $a\in A$, then $a\in B$ as well by assumption ($A\subseteq B$), so $A\subseteq A\cap B$.

$\Leftarrow:$ We always have $A\cap B\subseteq B$.

Since you are proving an "if and only if statement" you have two things to prove. Do you know what they are? If so, then you will see that in one direction you need to prove that two sets are equal? Do you know how to do this? What about proving that one set is a subset of another?