Definition of Algebraic Function Field

Question

the definition of algebraic function field $\mathbf{F}$ over $\mathbf{K}$ that I know is the following:

There exists an element $x \in \mathbf{F}$, transcendental over $\mathbf{K}$, such that $\mathbf{K}(x) \subset \mathbf{F}$ is a finite extension.

I want to show that the latter implies that $\mathbf{K} \subset \mathbf{F}$ has a transcendence degree $1$ and is finitely generated.

Does this fact have a nice proof?

Thanks in advance.