# How to show the well-definedness of a lift of a continuous map via a covering map?

Given a topological space $$X$$ is path-connected and locally path-connected. Say $$p:Z \rightarrow Y$$ is a covering map such that $$p(z_0)=y_0$$. A map $$f:X \rightarrow Y$$ is continuous and $$x_0 \in X$$ be such that $$f(x_0)=y_0$$. Also given that $$f_*(\pi_1(X,x_0))\subset p_*(\pi_1(Z,z_0))$$.

I am trying to show that $$f$$ can be lifted to a $$Z$$ via $$p$$ and the lift maps $$x_0$$ to $$z_0$$.

I have proceeded like this:

As $$X$$ is path connected so given $$x \in X$$, we have a path $$\alpha$$ from $$x_0$$ to $$x$$. Then I have a unique lifting of $$f \circ \alpha$$ to a path $$\widetilde{f \circ \alpha}$$ in $$Z$$ starting from $$z_0$$. I defined $$\tilde{f}$$ from $$X$$ to $$Z$$ as $$\tilde{f}(x)= \widetilde{f \circ \alpha}(1)$$.

I believe that $$\tilde{f}$$ will be the required lifting of $$f$$. And to show only the existence of such a lift local path-connectedness of $$X$$ is redundant. All it remains to finish the proof is that $$\tilde{f}$$ is well-defined i.e. independent of the path $$\alpha$$.

Am I proceeding correctly ? How do I show the well-definedness of $$\tilde{f}$$?