POTENTIAL CONFLICT AND POSSIBLE SYS ERROR WHEN AN ARR S80 CROSSES THE PATH OF A CLBING; DEPARTING COMMUTER FLT 34 NM NW OF CQY; TX.

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WE WERE ON THE CQY5 ARR TO DFW. WE HAD JUST CROSSED TACKE. WE WERE RETURNING FROM ATL TO DFW ON THE CQY5 ARR; LEVEL AT 11000 FT. WE GOT A TA CLOSE IN UNDER THE NOSE OF THE ACFT AFTER XING TACKE. I SAW AN ACFT AT ABOUT 1 1/2 - 2 MI LOW ON THE NOSE AT 90 DEGS TO OUR FLT PATH THAT APPEARED TO BE CLBING. AT THE SAME TIME; WE HAD AN RA THAT DIRECTED A HARD CLB. WE STARTED A HARD CLB TO THE GREEN ARC; THE LAST 3 LIGHTS ON THE ARC. I NOTIFIED APCH THAT WE WERE RESPONDING TO AN RA. WE CLBED TO 13000 FT BEFORE WE WERE CLR. WE THEN DSNDED BACK TO 11000 FT; AND MADE OUR TURN AT DIETZ. APCH AND LNDG WERE NORMAL. NO ONE WAS HURT BY THE RAPID CLB. MY FO WAS FLYING THE ACFT; AND DID AN EXCELLENT JOB IN RESPONDING TO THE RA. SUPPLEMENTAL INFO FROM ACN 578008: AS I STARTED THE PULL; WE RECEIVED INCREASE CLB WARNING. THE IVSI WAS COMPLETELY RED LIGHTS; EXCEPT FOR THE LAST 3 BEING GREEN. TCASII THEN SAID CONFLICT RESOLVED; AND I DSNDED BACK TO 11000 FT AND BEGAN SLOWING FOR DIETZ AT 210 KIAS. THE CAPT SAW THE CONFLICT ACFT AFTER I WAS WELL INTO THE PULL. THE SUN HAD SET; AND IT WAS TWILIGHT AT THE TIME. UPON LNDG DFW; GND CTL PASSED US A PHONE NUMBER TO CALL; AND THE CAPT CALLED AND TALKED TO THE APCH CTLR. NO FURTHER ACTION WAS REQUIRED. THE SAFETY COMMITTEE RECOMMENDED THIS RPT. I FEEL THAT THE TCASII WORKED AS ADVERTISED AND PREVENTED A MISHAP.