The chemical reactions and pathways resulting in the breakdown of a mitochondrial protein. This process is necessary to maintain the healthy state of mitochondria and is thought to occur via the induction of an intramitochondrial lysosome-like organelle that acts to eliminate the damaged oxidised mitochondrial proteins without destroying the mitochondrial structure. [ http://www.ncbi.nlm.nih.gov/pubmed/21264228 http://www.ncbi.nlm.nih.gov/pubmed/21264221 GOC:sp ]