Hall's theorem for bipartite graphs

Statement

Let $G$ be a finite bipartite graph with parts $X$ and $Y$. For $W \subseteq X$, let $N(W)$ be the set of vertices of $Y$ that are adjacent to some vertex of $W$. Then $G$ has a matching that covers every vertex of $X$ if and only if $|N(W)| \ge |W|$ for every $W \subseteq X$.

Remarks

If $|X| = |Y|$, such a matching is a perfect matching. Thus the condition then characterizes the perfect matchings. A consequence: a finite regular bipartite graph with at least one edge has a perfect matching. The theorem is the graph form of Hall's marriage theorem: let $X$ index the sets and let $Y$ be their union.

Explanations

L1 human

Think of $X$ as workers and $Y$ as jobs, with an edge when a worker can do a job. Every worker can get a different job exactly when this holds. Each group of workers can do, together, at least as many jobs as there are workers in the group.