Showing $1$-reducibility of $\overline{\text{HALT}'}$ to index set
Let $e$ be and index for the constant 0 function. Then consider $s$ such that

$$ \varphi_{s(x)}(y) = \begin{cases} 0 &amp; \mbox{ if }\varphi_x(x) \mbox{ does not halt within $y$ steps} \\ 1 &amp; \mbox{otherwise} \end{cases} $$

Then $\varphi_{s(x)} = \varphi_e$ if and only if $x \not\in HALT$ (assuming HALT was the diagonal halting problem).

The function $s$ can be choosed to be monotonic (use pumping), hence it is one-to-one.