Assuming we have 2 independent random variables, $X$ and $Y$. What is the probability event $A$ that corresponds to the following integration?
$\Pr (A) = \int_{-\infty}^{t} f_{Y} (y) F_{X}(y) dy$
where $F_{X}$ is the CDF of $X$ and $f_{Y}$ is the PDF of $Y$
If in this situation $Y$ has a PDF then:
$$\begin{aligned}P\left(X\leq Y\leq t\right) & =\int f_{Y}\left(y\right)P\left(X\leq Y\leq t\mid Y=y\right)dy\\ & =\int f_{Y}\left(y\right)P\left(X\leq y\leq t\right)dy\\ & =\int_{-\infty}^{t}f_{Y}\left(y\right)P\left(X\leq y\right)dy\\ & =\int_{-\infty}^{t}f_{Y}\left(y\right)F_{X}\left(y\right)dy \end{aligned} $$
The second equality is based on independence of $X$ and $Y$.