In many books and derivations of the Black-Scholes PDE one sees that
$$\Pi=V-\Delta F \Rightarrow d\Pi=dV-\Delta dF$$
which implicitly assumes that $d\Delta=0$. Somewhere down the road one then deduces that
$$\Delta=\frac{\partial V}{\partial F}$$
to simplify the equation. Doesn't this contradict the initial assumption that $d\Delta=0$? If one performs a full differentiation
$$\Pi=V-\Delta F \Rightarrow d\Pi=dV-\Delta dF - F d\Delta$$
the rest of the story goes wrong. Isn't it true that $\Delta = \Delta(t, S)$, i.e. is depending on time and the underlying stochastic process and hence has to be differentiated?