Not to nitpick, but the "stable" part isn't necessarily true. If the money is just being used for a down payment, and not being cited as recurring income, they wouldn't care whether it's stable.
Side note: I still can't make sense of why lenders care about the money's source being legitimate but the title company doesn't, per my experience: https://news.ycombinator.com/item?id=28163885
Well, I was conflating two things for simplicity’s sake.
They care that the down payment came from legitimate savings with a documented history, to ensure it’s not from another loan they’re not aware of.
And they care that your reported income is stable (you’re reliably employed with a reasonably predictable income stream) so they can reasonably expect you’ll continue to be able to make monthly payments.
Side note: I still can't make sense of why lenders care about the money's source being legitimate but the title company doesn't, per my experience: https://news.ycombinator.com/item?id=28163885