Birational maps and birational equivalence of classical varieties

Assume AC for the function-field and affine-atlas interfaces used here. For affine varieties, a dominant rational map $\mathrm{\Phi }:X⇢Y$ is birational if there is a dominant rational map $\mathrm{\Psi }:Y⇢X$ with $\mathrm{\Psi }\circ \mathrm{\Phi }={\mathrm{id}}_X$ and $\mathrm{\Phi }\circ \mathrm{\Psi }={\mathrm{id}}_Y$ as rational-map classes. Composition here is Dominant rational maps compose on nonempty open domains, the proved composition on nonempty inverse-image domains. For integral classical varieties equipped with compatible affine atlases, birational equivalence means the existence of isomorphic nonempty open subsets. The following theorem identifies this with the inverse-rational-map definition in the affine case and with a $k$-isomorphism of their function fields.

Source comparison: Milne, Algebraic Geometry, v6.10, §5l p. 117. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.