Implicit Function Theorem

Hey everyone I have a question on the proof of Implicit Function Theorem. My book says, "Suppose that z is given implicitly as a function z = f(x,y) by an equation of the form F(x,y,z) = 0. This means that F(x,y,f(x,y)) = 0 for all (x,y) in the domain of f. If F and f are differentiable, then we can use the Chain Rule to differentiate the equation F(x,y,z) = 0 as follows:

$\displaystyle \frac{\partial F}{\partial x}\frac{\partial x}{\partial x} + \frac{\partial F}{\partial y}\frac{\partial y}{\partial x} + \frac{\partial F}{\partial z}\frac{\partial z}{\partial x}=0$

But $\displaystyle \frac{\partial}{\partial x}(x) = 1$ and $\displaystyle \frac{\partial}{\partial x}(y) = 0$

So the equation becomes

$\displaystyle \frac{\partial F}{\partial x} + \frac{\partial F}{\partial z}\frac{\partial z}{\partial x}=0$

My question is, how does $\displaystyle \frac{\partial}{\partial x}(y) = 0$? It doesn't make sense. Is it just saying that let/assume $\displaystyle \frac{\partial}{\partial x}(y) = 0$? Confuseddd.

Thanks in advance!