This is a strengthening of the Knaster–Tarski theorem sometimes referred to as "Pataraia’s theorem". For example, see Section 4.1 of "Realizability at Work: Separating Two Constructive Notions of Finiteness" (2016) by Bezem et al. See also Chapter 4 of The foundations of program verification (1987), 2nd edition, Jacques Loeckx and Kurt Sieber, John Wiley & Sons, ISBN0-471-91282-4, where the Knaster–Tarski theorem, formulated over pointed dcpo's, is given to prove as exercise 4.3-5 on page 90.