Let $f:X \to Y$ be a regular map of quasi-projective varieties. If $x$ is a nonsingular point of $X$ does that imply $f(x)$ is a nonsingular point of $Y.$ Is the result true if $f$ were a regular isomorphism?
My attempt: If $f$ is a regular isomorphism the $df_x: \Theta_x\to \Theta_y$ is isomorphism and since $dim X= dim\Theta_x$ implies $dim Y= dim \Theta_y.$ Is my argument correct? is the result true if $f$ were just an regular map of quasi-projective varieties.