lifting theorem

Let p:EB be a covering map and f:XB be a (continuousMathworldPlanetmath) map where X, B and E are path connected and locally path connected ( Also let xX and eE be points such that f(x)=p(e). Then f lifts to a map f~:XE with f~(x)=e if and only if π1(f) maps π1(X,x) inside the image π1(p)(π1(E,e)), where π1 denotes the fundamental group functor. Furthermore f~ is unique (provided it exists of course).

The following diagrams might be useful: To check