Is the sole of a woman's foot considered 'awrah (parts of the body that must be covered), and is her prayer valid if it is uncovered?

For non-mahram men, a woman is entirely 'awrah (private, to be covered). However, for women and mahram relatives, the foot is not 'awrah. In prayer, she must cover the top and bottom of her feet. Nevertheless, exposing the sole of the foot does not invalidate the prayer according to the preferred view, even among those who deem its covering obligatory, because it is considered a minor part of the 'awrah.