Pistis Christou and the Law

I'm reading through Galatians as I work my way through Frank Thielman's Paul and the Law. And I have been overwhelmed by Paul's logic in Galatians 3 in regard to Christ and the Law. In verse 22, Paul writes that everything was "locked up" under sin so that the promise could be given "through faith in Jesus Christ . . . to those who believe." I am well aware that the debate surrounding the various constructions and uses of pistis Christou as objective or subjective genitive is far from settled at this juncture, it has at least provided the impetus for me to look with new eyes at this passage, and to discover what is plainly there with regard to Paul's logic for why the law is in place (and in fact the reading doesn't seem to change substantively with subjective or objective renderings of the genitive construction). Paul seems to be saying that the law isn't opposed to the promise, but is in fact the necessary precondition of, and instrument through which Christ's faithfulness accomplishes our justification. The law provides in a sense the mechanism through which justification on our behalf by Christ occurs, as Christ's faithfulness under the law yields forth in God's abundant gift of life. The law is thus fulfilled and the promise is then extended to all people without the the constraint of the law, which has fulfilled its purpose in Christ. In a sense, instead of thinking of Christ as circumventing the law to bring salvation by faith (a type of theology that may at times rest just beneath the surface in my own thinking), Christ brings that salvation precisely through the law, which was God's ordained mechanism to bring blessing and life from death, even as the gift is now offered apart from that same law.