Suppose $A,B,X$ are independent and disjoint sets of vertices in a graph such that $A \cup B \cup X = V$, $|A|=|B|=9$ and $|X| = 63$. Also, assume $d(v) = 7$ for every $v \in A,B$ and suppose that for every $x \in X$, $x$ is incident to exactly one $a \in A$ and one $b \in B$. Prove that there exists a set $Y \subseteq X$ such that for every $v \in A \cup B$, there exists $y \in Y$ such that $y$ is incident to $v$, and $|Y|=9$.
I tried to solve using the pigeonhole principle but the numbers didn't work. Any ideas?