TCASII RA TFC VERY CLOSE. CENTER CALLED TFC.

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WE WERE ON THE BLITHO 5 ARR TO KMCO AND WAS CLRED TO CROSS LAMMA INTXN AT 12000 FT AND 250 KTS. AT OMN 176 DEG/05 WE WERE DSNDING THROUGH 17000 FT AT 300 KTS AND APPROX 2000 FPM DSCNT WHEN OUR TCASII SOUNDED 'TFC.' WE THEN NOTICED THE TFC ON OUR SCREEN FOR THE FIRST TIME AT OUR 12 O'CLOCK POS APPROX 3 NM RANGE. MY SCOPE WAS SET AT THE 10 MI RANGE AND THIS TFC DID NOT SHOW UP PRIOR TO THIS. SIMULTANEOUSLY WITH THE TCASII TA JAX CENTER ADVISED US OF TFC AT OUR 12 O'CLOCK POS AND UNCONFIRMED ALT OF 16600 FT. WITHIN A SECOND OR 2 OUR TCASII CALLED 'DSND; DSND NOW' ALONG WITH A 2500 FPM MIN DSCNT RATE COMMAND. OUR ALT WAS THEN GOING THROUGH 16600 FT AND MY COPLT WHO WAS FLYING INITIATED A DIVING R TURN; I LAST NOTICED THE TFC ON OUR TCASII SCREEN AT LESS THAN A MI AND AT OUR ALT. ALL THIS TOOK PLACE IN WHAT SEEMED LIKE 5 OR 6 SECONDS. THIS RESULTED IN A VERY UNCOMFORTABLE DIVING R TURN AT 4000 FPM RATE OF DSCNT. I NEVER DID SEE THE TFC VISUALLY. UPON ARR AT KMCO I CALLED JAX CENTER AND THEY CONFIRMED THAT THERE WAS INDEED AN SMA FLYING VFR AT THAT POS. THE SMA HAD TO HAVE HAD AN OPERATING TRANSPONDER WITH ALT ENCODER TO SHOW UP ON OUR TCASII SCREEN AS IT DID. WHY DIDN'T THE TCASII ALERT SOUND EARLIER? WHY DIDN'T WE GET AN EARLIER ADVISORY FROM CENTER? PRIOR TO AND SUBSEQUENT TO THIS EVENT OUR TCASII SYS OPERATED PROPERLY. DID THE SMA JUST THEN TURN ON HIS TRANSPONDER? WHATEVER THE REASON; TCASII DOES WORK AND HAS FOUND ANOTHER BELIEVER IN THE SYS.