Inclusion and intersection equal to the subset

Statement

Let $A$ and $B$ be sets. Prove that $B \subseteq A$ if and only if $A \cap B = B$.

Solution

This is an “if and only if”: we prove both implications.

($\Rightarrow$) Assume $B \subseteq A$. We already know $A \cap B \subseteq B$. Conversely, let $x \in B$: since $B \subseteq A$ we also have $x \in A$, hence $x \in A \cap B$. So $B \subseteq A \cap B$. From the two inclusions, $A \cap B = B$.

($\Leftarrow$) Assume $A \cap B = B$. If $x \in B$, then $x \in A \cap B$, so in particular $x \in A$. Therefore $B \subseteq A$. $\blacksquare$