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?