Let $X,Y,Z$ be three disjoint subsets of events such that $\chi = X \cup Y \cup Z$. I'm trying to show that $X$ and $Y$ are independent given $Z$ if and only if there exist functions $\phi_1, \phi_2$ such that $\Pr(\chi) = \phi_1(X, Z) \phi_2(Y,Z)$.
I think I can do one direction: Assuming the independence, we have
$$\frac{\Pr(X,Y,Z)}{P(Y,Z)} = \Pr(X|Y,Z) = \Pr(X|Z) = \frac{\Pr(X,Z)}{\Pr(Z)} $$
so that $$\Pr(X,Y,Z) = \frac{\Pr(Y,Z)\Pr(X,Z)}{\Pr(Z)}.$$
So we can take, say, $\phi_1(X,Z) = \frac{\Pr(X,Z)}{\Pr(Z)}$ and $\phi_2(Y,Z) = \Pr(Y,Z)$. Right?
I'm having trouble with the other direction. If $\Pr(X,Y,Z)=\phi_1(X, Z) \phi_2(Y,Z)$ then
$$\Pr(X|Y,Z) = \frac{\Pr(X,Y,Z)}{\Pr(Y,Z)} = \frac{\phi_1(X, Z) \phi_2(Y,Z)}{\Pr(Y,Z)}.$$ I'm not sure what to do with this...
Actually, if X and Y are disjoint. Then they are definitely $\color{red}{dependent}$. The reason is that if X happens, then Y will never happen. Maybe you should change your hypothesis about $X$, $Y$,and $Z$.