functions - Proving: $f$ is injective $Leftrightarrow f(X cap Y) = f(X) cap f(Y)$

Let $A$ := "$f$ is injective" and $B$ := "$f(X \cap Y) = f(X) \cap f(Y)$".

My first idea is to show $B \implies A$ through contraposition, so $\lnot A \implies \lnot B$. Would it then be enough if I say: $f$ is not injective and then show an example where the equation in $B$ is wrong? Would it be a proof then?