Suppose I have $X,Y$, which are continuous random variables, with a joint PDF: $$ f_{X,Y}(x,y)= \begin{cases} 24xy & 0 \leq x \text{ and } 0 \leq y \text{ and } x+y \leq 1 \\ 0 & \text{otherwise} \\ \end{cases} $$
I want to find the joint CDF, which I have calculated to be $6x^2y^2 $. However, I'm not sure how to define the piecewise function for the joint CDF.
I wrote out the CDF below, but I'm not sure if the cases of $x$ and $y$ are correct for all $x,y \in \mathbb{R}$. $$F_{X,Y}(x,y)= \begin{cases} 0 & x < 0 \text { or } y < 0\\ 6x^2y^2 & 0 \leq x \leq 1 \text{ and } 0 \leq y \leq 1 \text{ and } x+y \leq 1\\ 1 & x > 0 \text{ and } y > 0 \text{ and } x+y > 1\\ \end{cases}$$
Look at the following drawing:
Thus the CDF is the following:
$$F_{XY}(x,y)=6x^2y^2\mathbb{1}_{(0;1)}(x)\mathbb{1}_{(0;1-x)}(y)+\mathbb{1}_{(0;1)}(x)\mathbb{1}_{(1-x;\infty)}(y)+\mathbb{1}_{(1;\infty)}(x)\mathbb{1}_{(0;\infty)}(y)$$