Independence of variables

Question

I have an odd question. Suppose $z$ is a function of $x,y$. That is, $z$ is dependent on $x,y$. This can be expressed by writing $z=f(x,y)$. This can in turn be written as $F(x,y,z)=0$, where $F(x,y,z)=f(x,y)-z$. My question is, can we call $F$ a function of three independent variables $x,y,z$?. On one hand it feels wrong because I know that $z$ depends on $x,y$. On the other hand, I can always define the function $F$ in the above manner regardless of any existing relationship among the three variables. Where am I thinking wrong? Should I use another variable to define $F$ instead of $z$?