Prove that $A\subseteq B\Longleftrightarrow A\cap B = A$ In set theory logic mathematics. How would i do the proof for: $A\subseteq B\Longleftrightarrow A\cap B = A$

First suppose that $A\subseteq B$.


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*Let $x\in A \cap B \implies x\in A $
So, $A\cap B\subseteq A$  

*Now, let $x\in A $. Since $A\subseteq B, x\in A\cap B \implies A\subseteq A\cap B$
Since $A\subseteq A\cap B$ and $A\cap B \subseteq A, A\cap B = A$

Now suppose that $A\cap B = A$
Let $x \in A \implies x \in A \cap B \implies x \in B$
Since $x \in A \implies x \in B, A\subseteq B$   

$$\therefore A\subseteq B\iff A\cap B=A$$
