Consider the set of indices of all total computable functions R = $\{m|\phi_{m}(x)\downarrow \forall x\in\mathbb{N}\}$.
Is $R\subset K=\{x|\phi_{x}(x)\downarrow\}$?
My argument is that when $\phi_{m}$ is total, $\phi_{m}(m)\downarrow$ and hence $m\in R$ and $m \in K$. Am I wrong?