CPR X CLB THROUGH OCCUPIED ALT HAD LTSS FROM Y. SYS ERROR.

Narrative

WHILE X WAS CLBING TO 4000 FT I MADE A POINTOUT TO ABE APCH FOR APPROVAL TO CLB TO 8000 FT; WHICH WAS APPROVED WITH A PROVISION ON TFC TO THE W; WHICH I BELIEVED WAS TFC (XING W TO E) AT 5000 FT. I THEN CLBED THE FALCON X TO 5000 FT TO KEEP HIM ON A CONTINUOUS CLB; AND CALLED HAR APCH FOR APPROVAL TO 8000 FT IN THE ABE APPROVED POINTOUT AREA. I WAS TOLD TO STAND BY. BECAUSE OF THE CONFLICTING ALTS OF THE DEPARTING AND XING TFC; I TURNED THE FALCON X TO 360 DEGS AND 5000 FT. HAR APCH HAD SINCE TAKEN THE AUTOMATED HDOF; SO I RECALLED THEM TO CLB TO 8000 FT; WHICH WAS THEN APPROVED. THE FALCON X WAS ISSUED THE CLB TO 8000 FT AND INSTRUCTED TO PROCEED (TURN R; NW) ON COURSE LEAVING 6000 FT OF THE TFC AT 5000 FT EBOUND. SHORTLY THEREAFTER; THE FALCON X RPTED A TCASII ALERT AND QUESTIONED POSSIBLE TFC AT 7000 FT AND AT THE SAME TIME HAR APCH CALLED TO INQUIRE IF I KNEW ABOUT Y FLT AT 7000 FT. I REPLIED; 'NEGATIVE' AND THAT THE FALCON WAS PROCEEDING ON COURSE. SUBSEQUENTLY; THE FALCON RPTED TCASII TFC AT 5 O'CLOCK AND 7000 FT AND THAT HE WAS LEAVING 6900 FT. I OBSERVED THE FALCON TO BE APPROX 4 MI FROM SAID TFC. CONTRIBUTING FACTORS: IT APPEARS THAT THE TFC BELIEVED TO BE A FACTOR FOR X WAS COLOCATED WITH AN ACFT THAT ACTUALLY WAS THE 7000 FT; ACFT Y; BUT SINCE I ASSUMED THAT THE TAGGED ACFT THAT ABE AND RDU WERE VIEWING WAS Y THE CLB WAS BASED ON. THE COORD WAS NOT THOROUGH.