B757-200 CREW AND INTRUDING ACFT HAD AN NMAC JUST AFTER TKOF IN DFW CLASS B.

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NEAR MIDAIR COLLISION. DEPARTING DFW ON RWY 18L; WE WERE A LIGHTLY LOADED 170000 LBS; B757 CAPT FLYING; CLBING RAPIDLY AFTER A STANDARD PWR TKOF AND CLB 1 SELECTED. WE WERE FLYING AT JUST OVER CLEAN MINIMUM MANEUVER AND ACCELERATING SLOWLY DUE TO BEING FAIRLY CLOSE TO THE TFC WHICH HAD DEPARTED IN FRONT OF US. AT ABOUT 3000-3500 FT; TCASII ANNOUNCED 'TFC; TFC.' I LOOKED DOWN AT THE SCREEN; WE HAD A YELLOW SYMBOL; OR MAYBE IT TURNED RED RIGHT THEN; I'M NOT CERTAIN; 900 FT ABOVE. I LOWERED THE NOSE SLIGHTLY DUE TO THE PROX OF THE OTHER ACFT AND IMMEDIATELY GOT A 'MONITOR VERT SPD.' THE GREEN ARC WAS AT THE TOP OF THE CLB SCALE AT ABOUT 3500-4000 FPM AND HAD JUST 3 GREEN LIGHTS LIT. ALL THE REST OF THE VSI WAS RED. AND; OF COURSE; THE CONFLICT ACFT SYMBOL WAS RED. THE TCASII THEN ANNOUNCED 'CLB XING CLB.' I PITCHED UP TO CLB AT THE RATE INDICATED BY THE GREEN LIGHTS; SELECTED 5 MI SCALE ON THE TCASII AND SAW THE SYMBOL AT 200 FT ABOVE US AND ACTUALLY OVERLAPPING THE WINGTIP OF OUR ACFT SYMBOL ON THE TCASII SCREEN. I LOOKED OUTSIDE; FOUND THE OTHER ACFT; LOWERED THE NOSE AND PROVIDED VISUAL CLRNC. THE OTHER ACFT THEN TURNED AWAY TOWARD THE E. THERE IS NO DOUBT THAT HAD I CONTINUED TO FOLLOW THE RA I WOULD HAVE FLOWN THROUGH THE OTHER ACFT'S VERT FLT PATH. IF WE HAD THEN MISSED THAT ACFT; IT WOULD HAVE BEEN ONLY LATERALLY AND BY AN UNCOMFORTABLY SMALL MARGIN. IT IS POSSIBLE THAT WE WOULD HAVE COLLIDED. THE TCASII DID A GOOD JOB OF ALERTING US TO THE PRESENCE OF THE OTHER ACFT; BUT; FROM OUR PERSPECTIVE; A NEARLY LETHAL JOB OF RESOLVING THE CONFLICT. I WAS SURPRISED TO SEE THE CLB COMMAND ON THE RA. CLRLY; A LEVELOFF OR A REDUCED RATE OF CLB FOR US WOULD HAVE RESOLVED THE CONFLICT EASILY WITH MUCH GREATER DISTANCE BTWN THE 2 ACFT. I CALLED THE TWR AND ASKED WHAT HAD HAPPENED. THE TWR CTLR SAID HE DIDN'T KNOW BUT THAT IT MUST HAVE BEEN AN ACFT FROM THE E COMPLEX THAT HAD DRIFTED OVER THE W SIDE. HE TOLD US TO TURN R TO 190 DEGS AND CONTACT DEP CTL.