Here "name" is a metonym for the power of God. Here "Father" is an important title for God. Jesus performed miracles through his Father's power and authority. Alternate translation: "through my Father's power" or "with my Father's power" (See: [[rc://en/ta/man/translate/figs-metonymy]] and [[rc://en/ta/man/translate/guidelines-sonofgodprinciples]])
His miracles offer proof about him like a person who testifies would offer proof in a court of law. Alternate translation: "offer proof concerning me" (See: [[rc://en/ta/man/translate/figs-personification]])