In Galatians 5:22, I think the idea is faithfulness.
In Romans 5:1-2, we have access to that grace by faith. Ephesians 2:8 says the same thing that we are saved by grace, through faith. In other words, faith takes u to that grace. Galatians 3:3 Paul asks the question, did we receive the Holy Spirit by the works of the Law or the hearing of faith? The answer is that we received the Spirit by faith. Receiving the Spirit is how we are raised up with Him and given new life per Ephesians 2:1-2. Vs. 5-6 says this about that new life - "even when we were dead in trespasses, made us alive together with Christ (by grace you have been saved), and raised us up together, and made us sit together in the heavenly places in Christ Jesus,". This placing into, or baptism, implied in Ephesians 2:1-2, is spoken about further and is the result of faith as Paul tells us in Galatians 3:22. But the Scripture has confined all under sin, that the promise by faith in Jesus Christ might be given to those who believe. And a bit later, he says this 26 For you are all sons of God through faith in Christ Jesus. 27 For as many of you as were baptized into Christ have put on Christ."
The life comes as a result of faith. What God does before that can still be a question, but the new birth is the result of faith, because the new birth comes from being immersed into oneness with Jesus, placed into Christ. As He and the Father are one, we are also one with Him when we receive the Holy Spirit indwelling, called the 'placing into with the Holy Spirit', or the 'immersion with the Holy Spirit', or the 'baptism with the Holy Spirit'. That's what gives us the life. The baptism, the receiving of the indwelling of the Holy Spirit is always a result of faith. Also consider Eph 2:1-2 with these passages -- 1 Peter 3:21, Romans 6:3-11, Col. 2:10-14.