 Uniqueness of lifts to a covering space

Let $p:\tilde{X}\to X$ be a covering map and let $Y$ be path-connected. If two lifts ${\tilde{f}}_1,{\tilde{f}}_2:Y\to \tilde{X}$ of the same map $f:Y\to X$ agree at one point, they agree everywhere. Join that point to any other by a path and apply uniqueness in the path lifting theorem.