By definition of $F^-$, for any $t \in \mathbb{R}$ and $u \in (0,1)$, the inequality $F^-(u) \leq t$ holds if and only if $u \leq F(t)$. Therefore $\mathbb{P}(X \leq t) = \mathbb{P}(F^-(U) \leq t) = \mathbb{P}(U \leq F(t)) = F(t)$, which is the cumulative distribution function of $P$.