Abstract interpretation offers sound and decidable approximations for undecidable queries related to program behavior. The effectiveness of an abstract interpretation process relies entirely on the abstract domain itself, and the worst-case scenario is when the abstract interpreter responds with ``don’t know", meaning that anything could happen during runtime. The concept of completeness relates to the answer precision level when performing computations within the abstract domain. However, completeness for a whole language is an ideal domain property, usually holding only on trivial situations [GiacobazziLR15]; for this reason, a local notion of completeness, holding on a specific program input, has been deeply investigated [BruniGGR21]. In this paper, we characterize an intermediate notion holding for sets of input selected by abstraction. In other words, completeness holds for a set of concrete inputs determined by one abstract input. In this sense, it is a form of local abstract completeness, being required locally on one specific abstract value. We provide a simple proof system for proving this weakening of completeness and several examples. Notably, this proof system is both language and domain-agnostic and can be readily incorporated to support static program analysis.