 Solution

For distinct $x,x^{\prime }\in X$ and distinct $y,y^{\prime }\in Y$, take all clausesThe first family says that each $x$ is paired with at most one $y$; the second says that each $y$ is paired with at most one $x$. There is no existence clause, so the domain may be any subset of $X$. Thus the relations $f_v$ are exactly the injective partial functions from $X$ to $Y$.