Problem
Let $M$ and $N$ be smooth manifolds and let $F: M\to N$ be any map. Then prove that the followings are equivalent,
For every $p \in M$ there exists smooth charts $(U,\varphi)$ containing $p$ and $(V,\psi)$ containing $F(p)$ such that $F(U)\subseteq V$ and the composite map $\psi\circ F\circ \varphi^{-1}$ is smooth from $\varphi(U)$ to $\psi(V)$.
For each $p\in M$ there exist smooth charts $(U,\varphi)$ containing $p$ and $(V,\psi)$ containing $F(p)$ such that $U\cap F^{-1}(V)$ is open in $M$ and the composite map $\psi\circ F\circ \varphi^{-1}$ is smooth from $\varphi(U\cap F^{-1}(V))$ to $\psi(V)$,
My Attempt
I think that $(1)\implies (2)$ is easy. It is so because if we assume $(1)$ then for every $p\in M$, there exists smooth charts $(U,\varphi)$ containing $p$ and $(V,\psi)$ containing $F(p)$ such that $F(U)\subseteq V$. Now $$F(U)\subseteq V\implies F^{-1}(F(U))\subseteq F^{-1}(V)\implies U\subseteq F^{-1}(F(U))\subseteq F^{-1}(V)$$ and hence $U\cap F^{-1}(V)=U$, which is open in $M$. Thus we have shown that for every $p \in M$ there exists smooth charts $(U,\varphi)$ containing $p$ and $(V,\psi)$ containing $F(p)$ such that $U\cap F^{-1}(V)$ is open in $M$ and the composite map $\psi\circ F\circ \varphi^{-1}$ is smooth from $\varphi(U\cap F^{-1}(V))$ to $\psi(V)$. hence we are done.
Where I am stuck
To prove that $(2)\implies (1)$, I think it is enough to show the existence of a chart $(V,\psi)$ containing $F(p)$ such that $F(U)\subseteq V$ because then $U\cap F^{-1}(V)=U$. However, to prove this the problem I am facing is that the hypothesis of $(2)$ implies that $U\cap F^{-1}(V)\ne \emptyset$. But how can I use it to show that $F(U)\subseteq V$?
$(1) \Rightarrow (2)$ is correct. Hint for $(2) \Rightarrow (1)$: restrict the chart $U$ to $U':=U \cap F^{-1}(V)$. The important thing is that there is some chart, so you can use $U'$ instead of $U$, since an open subset of a chart is still a chart.