lifting theorem

Let $p:E\to B$ be a covering map and $f:X\to B$ be a (continuous) map where $X$, $B$ and $E$ are path connected and locally path connected (http://planetmath.org/LocallyConnected). Also let $x\in X$ and $e\in E$ be points such that $f(x)=p(e)$. Then $f$ lifts to a map $\tilde{f}:X\to E$ with $\tilde{f}(x)=e$ if and only if ${\pi }_1(f)$ maps ${\pi }_1(X,x)$ inside the image ${\pi }_1(p)\left({\pi }_1(E,e)\right)$, where ${\pi }_1$ denotes the fundamental group functor. Furthermore $\tilde{f}$ is unique (provided it exists of course).

The following diagrams might be useful: To check