For family of distributions $\mathcal{D}$, random variable $X$, its distribution function $F \in \mathcal{D}$, $c \in \mathbb{R}$, and $\epsilon \in [0,1]$, I'm trying to characterize: $$ \begin{align} \sup_{F\in\mathcal{D}} \{\mathbb{P}_F(X < c)\} := \sup_{F\in\mathcal{D}} \{F(c)\} &\leq \epsilon \end{align} $$ in terms of $X$'s quantile function $F^{-}(u) := \inf \{x: F(x) \geq u\}$. I have referenced these questions but am not sure how to use them.
What I want to do is say something like: $$ \begin{align} \sup_{F\in\mathcal{D}} \{ F(c) \} &\leq \epsilon \iff c \leq \inf_{F\in\mathcal{D}} \{ F^{-}(\epsilon) \} \end{align} $$ but I'm not sure if this is an equivalent characterization. Is this the right way to relate $\sup$ and $\inf$ with an inequality? Am I implicitly assuming some nice properties (continuity or monotonicity, maybe) about $F$?