Proof verification about joint distribution

Question

If $X$ and $Y$ are independent non-negative random variables, is the proof below correct? $\text{Pr} \left\lbrace X+Y \leq C \right\rbrace =\int\limits_{0}^{C} \, \int\limits_{0}^{C-x} f_{X}\left(x \right) f_{Y} \left(y \right) \, \text{d}y \, \text{d}x =\int\limits_{0}^{C} f_{X} \left( x \right) \int\limits_{0}^{C-x} f_{Y} \left( y \right) \, \text{d}y \, \text{d}x= \\ \int\limits_{0}^{C} f_{X} \left(x \right) \, F_{Y} \left( C-x \right) \text{d}x $