Let $\{\phi_i:U_i\to V_i\}$ be an atlas of manifold $M^m$. Tangent vector is a an equivalence class $[x,i,u]$ where, $x\in U_i, u\in\mathbb R^m$. Two classes $[x,i,u]$ and $[x',j',v]$ are equal if $x=x'$ and $(d_{\phi_i(x)}(\phi_j\circ\phi_i^{-1}))(u)=v$.
Atlas of tangent manifold $\{U_i'\}$ is defined as $U_i'=\{[x,i,u]:x\in U_i, u\in\mathbb R^n\}$ and charts $\phi_i':U_i'\to V_i\times\mathbb R^n$ as $\phi_i'([x,i,u])=(\phi_i(x),u)$. Can someone explain why is $\phi_i'$ well defined i.e. why is $\phi_i'([x,i,u])=\phi_i'([x',j,v])$ if $[x,i,u]=[x',j',v]$?
