Local and nonlocal implicit function theorems are obtained for closed mappings with a parameter from one Asplund space to another. These theorems are formulated in terms of the regular coderivative of a mapping at a point. The obtained results are applied to study properties of the minimum function for a constrained extremum problem with equality-type constraints and with a parameter. Sufficient conditions for the upper semicontinuity of the minimum function for a given parameter value are obtained.