In this paper, we study statistical inference of change-points (CPs) in multi-dimensional sequence. In CP detection from a multi-dimensional sequence, it is often desirable not only to detect the location, but also to identify the subset of the components in which the change occurs.Several algorithms have been proposed for such problems, but no valid exact inference method has been established to evaluate the statistical reliability of the detected locations and components. In this study, we propose a method that can guarantee the statistical reliability of both the location and the components of the detected changes. We demonstrate the effectiveness of the proposed method by applying it to the problems of genomic abnormality identification and human behavior analysis.