Per image below, how does A transform into B? I'm referring to specifically, how Px,y(x,z-x) turns into px(x)py(z-x) Thanks a lot for looking over my newbie question!
The problem statement is incomplete. The problem should have specified as an assumption that $X$ and $Y$ are independent. For independent random variables $X$ and $Y$, we have $$\Pr((X,Y)=(a,b))=\Pr(X=a)\Pr(Y=b).$$ That is what is being used to get from A to B.