Given a continuous map $f:M_{1}\rightarrow M_{2}$ between differentiable manifolds, a map is smooth if for all $p\in M_{1}$ with there exist charts $\varphi_{1}:U_{1}\rightarrow V_{1}$ and $\varphi_{2}:U_{2}\rightarrow V_{2}$ in $M_{1},M_{2}$ respectively (with $p\in U_{1},f(p)\in U_{2}$) such that the map $\varphi_{2}\circ f\circ\varphi_{1}^{-1}$ is a smooth map between Euclidean spaces.
Why is the smoothness of this map independent of the charts? This is clear to me if we change either of the $\phi_{i}$ or $V_{i}$ (immediate from compatibility), or if we make $U_{1}$ smaller, but I can't see why it should be true in general.