While I was reading a proof about the diagonalizability of symmetric matrices I got a bit lost the author was supposed to show that $f(H) \subseteq H$ but he ended up showing that
$y \in H \implies f(y) \in H$ for all $y$
claiming it's the same thing
I would say because $y$ is arbitrary here $f(y)$ is practically the same as $f(H)$ since $y \in H$ looking at it this way it makes some sense to me but I'm still confused why those two statements are equivalent.
Please could anybody here clearly explain why?