Existence of a lifting for simply connected covering. I want to prove the following statement:

Let $X$ be a path connected space, $p: E \to Y$ be a covering map, and
  $f: X \to Y $ be continuous. 
If $E$ is simply connected, and the image of $f_{*}(\pi_1(X))$ is
  nontrivial, then $\nexists$ a lifting $F : X \to E$ such that $p \circ
 F = f$

Am I supposed to prove it by contradiction? Is there any theorem that I can apply?

Let $F:X \to E$ be a lift of $f: X \to Y$ along $p : E \to Y$. The induced map on fundamental groups gives $f_* = p_* \circ F_* : \pi_1(X) \to \pi_1(E) \to \pi_1(Y)$. But $\pi_1(E)=0$ by assumption so $f_*$ must have a trivial image, contradiction.
