Of course local churches are correctly termed as “church” but it is wrong to use arithmetic language such as the term “subset” to refer to visible groups and say they attach to each other. Since the Invisible Church (Universal Church) is known only to God, no one can present evidence that visible churches subsume to the unseen group that God only sees.
Its almost like trying to use math in distinguishing “Not all Israel is Israel” (see Rom. 9.6-7). No, God foundation is immovable and sealed: He knows who are His (2Tim. 2.19).