Bipartite neighbourhoods, Hall's condition and systems of distinct representatives

Let $G$ be a finite bipartite graph with specified parts $X$ and $Y$. For $S\subseteq X$, its neighbourhood in $Y$ is $N(S):=\{y\in Y:xy\in E(G)\text{ for some }x\in S\}.$ The pair $(X,Y)$ satisfies Hall's condition (on $X$) if $|N(S)|\ge |S|$ for every $S\subseteq X$.

For any indexed family $(A_x)_{x\in X}$, write $U:=\bigcup_{x\in X}A_x$. A system of distinct representatives (SDR) is an injection $r:X\to U$ such that $r(x)\in A_x$ for every $x\in X$.

When $X$ and $U$ are finite, the incidence graph of the family is the finite bipartite graph with the disjoint tagged parts $X_{\mathrm L}:=\{(x,\mathrm L):x\in X\},\qquad U_{\mathrm R}:=\{(u,\mathrm R):u\in U\}.$ and an edge $(x,\mathrm L)(u,\mathrm R)$ exactly when $u\in A_x$. For $S\subseteq X$, its left tagged copy has neighbourhood $N(S_{\mathrm L})=\{(u,\mathrm R):u\in\bigcup_{x\in S}A_x\}.$