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 $N(S):=\{y\in Y:xy\in E(G)\text{ for some }x\in S\}\mathrm{.}$ The pair $(X,Y)$ satisfies Hall's condition (on $X$) if $\mathrm{\mid }N(S)\mathrm{\mid }\ge \mathrm{\mid }S\mathrm{\mid }$ 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 $X_{\mathrm{L}}:=\{(x,\mathrm{L}):x\in X\},U_{\mathrm{R}}:=\{(u,\mathrm{R}):u\in U\}\mathrm{.}$ 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 $N(S_{\mathrm{L}})=\{(u,\mathrm{R}):u\in {\bigcup }_{x\in S}{A}_x\}\mathrm{.}$