How do you prove that $A\cap B=B$ is equivalent to $B\subseteq A$?

Question

My question is regarding how to prove that $A\cap B=B\iff B\subseteq A$. I can understand it logically: let $x$ be an arbitrary element in set $B$. Since $A\cap B=B$, this means that $x$ must be an element of both $A$ and $B$, because it's in their intersection. Therefore, $x\in A$ and $x\in B$. But I don't know how to go further.

Answer 1

Let us prove first that $A\cap B = B$ implies that $B\subseteq A$. If $x\in B = A\cap B$, this implies that $x\in A$ and $x\in B$. In particular, $x\in A$. Consequently, the proposed result holds. Suppose now that $B\subseteq A$. If $x\in B$, then $x\in A$, that is to say $x\in A\cap B$. On the other hand, if $x\in A\cap B$, then $x\in A$ and $x\in B$. In particular, $x\in B$ and we are done.

Hopefully this helps!

Answer 2

Suppose first that $A\cap B=B$ and let $x\in B$. Then $x\in A\cap B$, i.e., in particular, $x\in A$.
Conversely, let $B\subset A$. Then $A\cap B=B$.\

In case you are wondering there is another iff condition that one can use: $A\cap B=B\iff B\subset A \iff A\triangle B=A\setminus B. $