This paper provides new Farkas-type results characterizing the inclusion A ⊂ B, where A and B are subsets of a locally convex space X. The sets A and B are described here by means of vector functions from X to other locally convex spaces Y (equipped with the partial ordering associated with a given convex cone K ⊂ Y) and Z, respectively. These new Farkas lemmas are obtained via the complete characterization of the K-epigraphs of certain conjugate mappings which constitute the core of our approach. In contrast with a previous paper of three of the autors [J. Optim. Theory Appl. 173, 357-390 (2017)], the characterizations of A ⊂ B are expressed here in terms of the data.

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