I'm reading John Lee's smooth manifolds. I have a simple question about the Proposition $8.16$. Sorry about the long text which is necessary to explain my question.
Here is the statement of the proposition and its proof.
In the statement, he says that "for every smooth real-valued function $f$ defined on $\color{red}{an}$ open subset". Let's denote this set(i.e. $dom(f)\,$) by $D$.
I think for this set $D$, we must have that $F(M)\subseteq D$. But I am not sure if my thought is right. So my question is, does $D$ must satisfy that $F(M)\subseteq D$?
My reasoning: If $F(M)\not\subseteq D$, then $\exists \,p$ s.t. $F(p)\notin D$. Hence, there exists an small neighborhood of $F(p)$ in which $f$ is not defined. Then the expression $dF_p (X_p)f$ makes no sense because $dF_p (X_p)\in T_{F(p)} N$ and $f$ is not defined in a neighborhood of $F(p)$.
I am really confused now and I've been thinking of this for an hour. Thanks for help.