Let $k$ be an algebraically closed field, $X,Y$ projective varieties (irreducible algebraic sets) and $f:X\to Y$ a morphism. Is $f(X)$ a projective variety? I think it is because the image of a morphism is closed and continuity preserves irreducibility. Is this correct?
I wonder because if $X$ and $Y$ are affine varieties, the statement is not true by this example: Image of a morphism of varieties.