Prove $B = \{ \varphi_n(n) | n \in \mathsf{K} \}$ to be recursive

Question

The set $B$ is the range of universal function given the domain $\mathsf{K}$, where $\mathsf{K} = \{ n | \varphi_n(n) \textit{ halts} \}$. How can we prove such claim?

Answer 1

You might find it easier to prove the stronger result that $B$ contains all of the natural numbers. (A sneaky question)