Consider $P_0$ and $P_1$ principal G-bundles with projection maps $\pi_0, \pi_1$, respectively; $f:P_0 \rightarrow P_1$ a continuous G-equivariant map (i.e. a bundle map) and $g:X_0 \rightarrow X_1$ the induced map such that $g(x)=\pi_1(f(v))$ for every $x \in X_0$ and $v\in \pi_0^{-1}(x)$. Then we say that $f$ covers the map $g$.
I want to prove that $f$ is a bundle isomorphism between $P_0$ and $P_1$ iff $g$ is a homeomorphism between $X_0$ and $X_1$.
I've already proved that if $f$ is an isomorphism, then the induced $g$ is a homeomorphism. However, given that $g$ is an homeomorphism, it's not clear to me how should I define $f^{-1}$ in order to prove that it is a well-defined continuous inverse for $f$ and that it's G-equivariant.
Any help would be appreciated. Thank you in advance.