Prove that $A \smallsetminus (A \smallsetminus B) = A \cap B$ $A$ and $B$ are any sets, prove that $A \smallsetminus (A \smallsetminus B) = A \cap B.$ This formula makes sense when represented on a Venn diagram, but I am having trouble with proving it mathematically.
I have tried letting $x$ be an element of $A$ and continue from there, but it doesn't seem to work out as a valid proof anyways. 
Could anyone please point me in the right direction?
Many thanks. 

Let $x \in A \setminus (A \setminus B).$ 
Then  $x$ is an element such that $x \in A$ and $x \notin A \setminus B$. But if $x \notin A\setminus B$, with some additional work, we realize this implies that $x \in A$ and $ x \in B$. So $x \in A \cap B$.
Vice-versa: let $x \in A \cap B$ so $x \in A$ and $x \in B$. This implies  that $x \notin A \setminus B$. But given that $x \in A$ and $x \notin A \setminus B$ this implies that $x \in A \setminus (A \setminus B)$.
