In the study of equilibria in dimension higher than a small number, one is overburdened by excessive riches. Namely, there are so many equilibria that one cannot even hope to find them all. Therefore we are destined to concentrate on individual branches of equilibrium solutions. While a rigorous theoretical understanding of the intricacies of these branches is ideal, this is in general not a tractable problem, and numerical bifurcation searches are useful to inform study. However, it is important to have a firm assurance not only that the structure is accurate, but also that we have neither missed something nor incorrectly surmised the existence of a branch of solutions when none exists. Along with Thomas Wanner, I have developed a method of computer-assisted proofs using the constructive implicit theorem to prove both the existence of and the isolation of branches of equilibrium solutions and their bifurcations. This method is quite general. To illustrate this, I will describe applications of these results for three different examples: An iterated map, which is an age-structured model for coral population model in 13 dimensions; The discrete Allen-Cahn differential equation, which we studied in 10 and 100 dimensions; and the Ohta-Kawasaki partial differential equation model for the dynamics of diblock copolymers in one, two, and three space dimensions.