# Is this a computable function? Is the reduction correct?

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?