Let $ A$ be a set, $ K=\{x:\phi_x(x)\downarrow\}$ . Let c to be a total computable function such that $ \phi_{c(x,y,n)}(z)=\begin{cases}\phi_n(z) & \text{if }\phi_x(y)\downarrow\\uparrow &\text{otherwise}\end{cases}$

Suppose $ \forall x,y\exists a.\phi_x(y)\downarrow \Leftrightarrow c(x, y,a)\in A$ .

The question is if the function: $ f(x)=a$ such that $ x\in K \Leftrightarrow c(x, x, a)\in A$ is total computable.

Hence, can I prove $ K\leq _m A$ with $ c(x,x, f(x))$ as reduction function?