Let $A, B, C$ be sets. Prove that if $B \subseteq C$, then $(A\cap B)\subseteq (A\cap C)$
I tried this question so I'll tell you guys what I have. Please correct me if I am wrong.
Assume $B \subseteq C$, then if $x$ is an element of $B$, $x$ must be an element of $C$.
Then if $A$ was added to set $B$ and $A$ was added to set $C$, every element in $A\cap B$ must belong to $A\cap C$.
I'm not really sure how to write that mathematically or symbolically since I'm new to this