Let $X$ and $Y$ be subsets of $\mathbb{R}$ and let $F:X\times Y\to \mathbb{R}^N$ be some $N$-valued function that satisfies the following property: there exists an $x'\in X$ and $y'\in Y$ such that $$ \left(F(x', y')-F(x,y')\right)\cdot z \geq 0 $$ for all $x<x'$ and some $z\in \mathbb{R}^N$, then $$ \left(F(x', y')-F(x',y)\right)\cdot z \geq 0 $$ for some $y<y'$.
Do you recognize what this property may be? It makes me think it is a modification of min-max properties as well as single-crossing/supermodularity properties. I am trying to understand if there is a deeper interpretation. I am sorry that I cannot make the question more pointed.