Dominant affine images contain a principal open

Given: The objects and hypotheses in the statement.

Let $f:X\to Y$ be dominant between irreducible affine varieties, put $A=k[Y]\subseteq B=k[X]$, and let $r={\mathrm{trdeg}}_{k(Y)}k(X)$. There are $0\mathrm{\ne }a\in A$ and elements $t_1,\dots ,t_r\in B_a$, algebraically independent over $A_a$, such that $B_a$ is module-finite over $A_a[t_1,\dots ,t_r]$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (A dominant affine map factors finitely over relative affine space after shrinking the base).

Assume the Axiom of Choice. Let $f:A\to B$ be an integral ring map, and let $\mathfrak{p}\in \mathrm{Spec}(A)$ with $\ker f\subseteq \mathfrak{p}$. Then there exists a prime ideal $\mathfrak{q}\in \mathrm{Spec}(B)$ such that $f^{-1}(\mathfrak{q})=\mathfrak{p}$. (Lying over for integral ring maps).

Assume the Axiom of Choice. Let $k$ be an algebraically closed field. 1. The assignments $X⟼I(X),J⟼V(J)$ induce mutually inverse inclusion-reversing correspondences between affine algebraic sets $X\subseteq {\mathbf{A}}_k^n$ and radical ideals $J\subseteq k[x_1,\dots ,x_n]$. 2. Under this correspondence, nonempty irreducible affine algebraic sets correspond exactly to prime ideals. (Affine algebraic sets correspond to radical ideals, and irreducible ones to prime ideals).