Let $U_1,...U_n$ be i.i.d. random variables of uniform distribution on $[0,1]$ and $X_1,...X_n$ i.i.d. real random variables with common cumulative distribution function (cdf) $F$ and empirical cdf $F_n(t)=\frac{1}{n}\sum_{i=1}^{n}\mathbb{1}_{X_i \leq t}$. Denote by $G_n$ the empirical cdf of $U_1,...U_n$. I don't understand why we have the following equalities in distribution :
$$F^{-1}\left(G_n^{-1}\right)\stackrel{\mathcal{D}}{=}F_n^{-1}$$ $$F_n^{-1}\left(F^{-1}\right)\stackrel{\mathcal{D}}{=}G_n$$ where $F^{-1}(u)=\inf \left\{ t \in \mathbb{R}, F(t) \geq u\right\}$ is the quantile function related to $F$, same for $F_n^{-1}$ and $G_n^{-1}$.