At some point the phrase 'House of Israel' came to refer to all Jews, but much earlier it referred only to the northern kingdom of Israel, and not Judah. How long after the fall of Israel (c.722 BCE) did people of the southern kingdom of Judah come to understand the phrase as referring to themselves, and how?
Reviewing the occurrences of the phrase 'House of Israel' suggests to me that the early northern prophets, Amos and Hosea, used the phrase in the old sense as we'd expect, meaning the northern kingdom, same as it did in the early Deuteronomistic history. Even a century after Israel's fall Jeremiah very often still used it parallel to 'House of Judah', so still distinct (Micah and Zechariah also only used it as a parallel to Jacob or Judah). But it seems Ezekiel (c.580) -- and only Ezekiel -- used the phrase (a lot!) when he clearly meant Judah.
Was it a consequence of the consolidation of the religion in Exile that Judahites took on the 'House of Israel' mantle for themselves? And at what point was Ezekiel's apparent comfort with the phrase assumed by Jews more generally?