Condition for perfect matching in bipartite graphs
Here is a proof for the following related result:


  A $d$-regular bipartite graph contains a perfect matching.


According to Hall's criterion, a perfect matching exists if for all $A \subseteq S$, $|N(A)| \geq |A|$. Take any $A \subseteq S$. There are $d|A|$ edges connecting vertices in $A$ to $T$. A given vertex $t \in T$ can be the target of at most $d$ such edges. Hence there are at least $d|A|/d = |A|$ such targets, that is, $|N(A)| \geq |A|$.

Try modifying this proof for your case.