Flight Instructor reported a NMAC during approach training with VFR opposite direction departure traffic. By taking an evasive action an airborne collision was avoided.

Narrative

Flying IFR into ZZZ via the ZZZZZX arrival while in contact with ZZZ Approach. After passing ZZZZZ1 Intersection (@9000MSL and <230KIAS) the arrival required a course of 117* for vectors to the airport. The controller vectored us to the left (I can't remember the actual heading). The controller then gave us a lower altitude and we started to descend. We have a Garmin Traffic avoidance system in our aircraft and during the descent we had a traffic alert for an opposite direction traffic 500 ft below us. I leveled off and that aircraft passed below us. I closed the alert and began to descend to my assigned altitude. I was adjusting my airspeed to maintain <230 kts. and at that point I was looking down to accomplish that and establish an appropriate descent configuration. The co-pilot was looking outside and he spotted the opposite direction traffic and took control of the aircraft and rolled the aircraft left. I looked up and saw the other aircraft in a left bank. The aircraft was a (Diamond DA42). I then took control of our aircraft and I told the co-pilot to report the near hit that just occurred and ask why he didn't give us the traffic. He said it was because it was 'VFR traffic'. I didn't press the discussion because I was going to call ZZZ and discuss this with the operations manager. We called the airport in sight and were told to proceed west to enter a left downwind for XXL and cleared for the visual. The controller then handed us off to the Tower and we landed on Runway XXL at ZZZ. I called ZZZ Approach and talked to the operations manager. Then went to the flight school and talked to the manager and the pilot of the other aircraft. The other pilot was instructing single engine operation in an area that is under and just inside 0f A-XXX That begins at 7000 and ends at 17;000 both MSL. The pilot said their SOP states that they must contact ZZZ Approach if they are above 9000 MSL. The sectional states to monitor XXX.XX when in the alert area. In conclusion:The controller failed to separate an IFR aircraft (us) from VFR traffic in the area; however he did give us the traffic that was 500 ft. below us.The Tower Controller at ZZZ departed the DA42 straight out when they should have known of our IFR arrival inbound which would be in opposite direction to the departing traffic. The near hit occurred approximately 12.5 nautical from ZZZ.ZZZZZ1 to the airport is inside A-XXX until the descent takes the aircraft below the alert area (<7000MSL) floor. The RNAV approach for XXR also begins at ZZZZZ1VFR aircraft should not be practicing VFR maneuvers inside this corridor The VFR sectional states all aircraft should be monitoring freq XXX.XX while in the alert area. It should be below it also and they should be talking to ZZZ Approach since they are inside the approach area to ZZZ