IN VISUAL CONDITIONS AFTER A TA WAS GIVEN AND TFC WAS IN SIGHT; AN ACR CREW STILL RECEIVED A TCASII RA.

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AFTER A NORMAL TKOF INTO VFR/DAWN CONDITIONS; WE WERE GIVEN A VECTOR TO FALLS VOR AND CLRNC TO CLB TO 13000 FT. PASSING 5000 FT; WE WERE TOLD TO LEVEL AT 7000 FT WITH TFC; A CESSNA; TRAVELING IN THE OPPOSITE DIRECTION. WE STATED WE ACQUIRED THE TFC AND WERE SUBSEQUENTLY GIVEN A CLB TO 13000 FT WITH INSTRUCTIONS TO AVOID OPPOSITE DIRECTION TFC. AS WE CONTINUED OUR CLB; WE COULD SEE THE ONCOMING TFC. IT WAS CLR THAT WE WOULD MISS EACH OTHER. TCASII GAVE A 'TFC; TFC' ALERT. I DID NOT ALTER THE ACFT'S DIRECTION OR ALT. UPON FURTHER CONVERGENCE; TCASII THEN STATED THE RA 'CLB; CLB.' STILL OBSERVING THE CESSNA; I FELT NO NEED TO BE OVERLY URGENT AS I SAW NO CONFLICT WITH THE CESSNA BUT DID CONTINUE THE CLB. SHORTLY THEREAFTER; THE SECOND RA ANNOUNCED 'INCREASE CLB; INCREASE CLB.' I WAS VISUAL WITH THE CESSNA; STILL SEEING NO CONFLICT; AND VERY SHORTLY THEREAFTER; AS THE 2 AIRPLANES PASSED; TCASII ANNOUNCED 'CLR OF CONFLICT.' OUR SEPARATION PROBABLY BECAUSE OF THE SMALL SIZE OF THE CESSNA WAS CLOSER TO 1/2 - 1 MI. MY FO FELT IT WASN'T TOO CLOSE AND NOBODY SAID ANYTHING -- NOT ATC; NOT THE CESSNA PLT; NOR I. I FEEL; WITH 2/02/ HINDSIGHT; THAT IT COULD HAVE BEEN DISASTROUS WITHIN A SPLIT SECOND. I SHOULD HAVE BEEN PROACTIVE WITH TCASII EVEN THOUGH I WAS VFR; IN THIS CASE INCREASING CLB. FATIGUE AS A FACTOR; THIS WAS MY 11TH OUT OF 15 DAYS; MOSTLY EARLY TKOFS.