The graph of a smooth map is an embedded submanifold

Consider the map $G:M\times N\to N\times N$ defined by $G(p,q)=(F(p),q)$. In product charts on $M\times N$ and $N\times N$, its representative has the form $(u,v)\mapsto (\widehat{F}(u),v)$, so $G$ is smooth. The graph satisfies ${\mathrm{\Gamma }}_F=G^{-1}({\mathrm{\Delta }}_N)$.

Fix $(p,F(p))\in {\mathrm{\Gamma }}_F$. Choose a chart $\psi :V\to \mathrm{\Omega }\subseteq {\mathbb{R}}^n$ on $N$ at $F(p)$. By [L3], the product chart $\psi \times \psi$ identifies a neighbourhood of $(F(p),F(p))$ in $N\times N$ with $\mathrm{\Omega }\times \mathrm{\Omega }$, and in these coordinates the diagonal from [L1] is the set $\{(a,b):a=b\}$. The difference map $D:\mathrm{\Omega }\times \mathrm{\Omega }\to {\mathbb{R}}^n$, $D(a,b)=b-a$, is a smooth submersion with zero fibre exactly that diagonal slice.

Choose a chart $\phi :U\to \mathrm{\Lambda }$ on $M$ at $p$ with $F(U)\subseteq V$, and define $H:=D\circ (\psi \times \psi )\circ G\circ (\phi \times \psi {)}^{-1}:\mathrm{\Lambda }\times \mathrm{\Omega }\to {\mathbb{R}}^n$. Then $H(u,v)=v-\widehat{F}(u)$, so the Jacobian of $H$ has block form $\left[\begin{array}{cc}-D\widehat{F}(u)& I_n\end{array}\right]$ and is surjective at every point. Its zero fibre is exactly the graph in these coordinates. Thus $0$ is a regular value of $H$.

By [L2], the zero fibre of $H$ is an embedded codimension-$n$ submanifold of $M\times N$ near $(p,F(p))$. Since $(p,F(p))$ was arbitrary, ${\mathrm{\Gamma }}_F$ is an embedded submanifold. Its ambient dimension is $\dim M+n$, so its dimension is $\dim M$.