Relationship between DTFT and z-transform

$ X(w) = F{x[n]} = \sum_{n=-\infty}^\infty x[n]e^{-jwn} $

$ X(z)|_{z=e^{jw}} = X(e^{jw}) $

Can compute Z-Transform as a DTFT write $ X(z)=X(re^{jw}) $

then $ X(z)= \sum_{-\infty}^\infty x[n]z^{-n} $

$ X(z)= \sum_{-\infty}^\infty x[n](re^{jw})^{-n} $

$ X(z)= \sum_{-\infty}^\infty x[n]r^{-n}e^{-jwn} $

$ = F{x[n]r^{-n}} $