So I'm trying to understand the proof on page 63
http://www.math.cornell.edu/~hatcher/AT/AT.pdf
In the proof he says that if $\tilde{f}_{1}(y) \not = \tilde{f}_2 (y)$, then $\tilde{U}_{1} \not = \tilde{U}_2$. But,how is this true? You can't deduce this from anything. Certainly, there must be something magical happening for him to deduce this. Surely, they the maps could be taken the points to different places in $\tilde{U}_1, \tilde{U}_2$, but they are still equal.