What’s the source for the fact that we consider anyone born to a Jewish mother or converted according to Halacha to be a Jew?
The source is from tractate Kiddushin 68b, where the sages received it as part of the oral law handed down from Moses from Sinai and extrapolated it from the verse in Deuteronomy 7-4, where the Torah discusses intermarriage. The torah says “Don’t intermarry, your daughter you should not give for their son and their son should not take your daughter.” Then the torah says that the reason is because the intermarried partner will cause your child (the one born out of the marriage to stray from the path of the torah). The torah however only says that HE (meaning the man that has married your Jewish daughter) will cause the child to stray. the torah doesn’t mention that if your son marries a gentile woman that she will cause the child to stray. the reason is because that child will not stray from the torah, because he isn’t Jewish. It is from here and from the tradition handed down from Moses that the deciding factor if a child is Jewish is dependent on the mother not the father.
Anyone who converts is a Jew, not in the merit of his or her parents, but in their own merit.