Let $M$ be a differentiable manifold and $F:M\to N$ be a bijective mapping. If $M$ is a topological space, then $N$ admits a unique topology making $F$ a homeomorphism; it is easy prove that this topology could be: $$T_N=\Big\{F(U):U\text{ is open in }M\Big\}$$ If $\mathfrak{A}_M=\{(U_\alpha,\varphi_\alpha)\}$ is a differentiable atlas in $M$, we can show that $\mathfrak{A}_N=\{(V_\alpha,\psi_\alpha)\}_{\alpha\in A}$ constitutes a differentiable atlas in $N$; with: $$V_\alpha = F(U_\alpha)\qquad\text{ and }\qquad \psi_\alpha=\varphi_\alpha \circ F^{-1}$$ for any $\alpha \in A$.
How I can prove that $\mathfrak{A}_N$ is the unique differentiable atlas on $N$ that makes $F$ diffeomorphism?
If $I$ suppose $(W,\phi)$ local chart in $N$ such that $V_\alpha\cap W\ne \phi$ for some $\alpha \in A$, and I prove that $W$ and $V$ are compatibles, this implies that $\mathfrak{A}_N$ is unique by this way?
Many thanks!