A CARJ CREW; DSNDING INTO DFW; RESPONDED TO A CLB RA FROM TCASII.

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PROB AROSE WHILE IN DSCNT INTO DFW. FO WAS CONTACTING COMPANY ON SEPARATE RADIO; WHILE WE WERE IN LEVEL FLT AT 11000 FT ON APCH INTO DFW FROM MEMPHIS. APCH CTLR CALLED MY FLT AND GAVE ME A DSCNT TO 5000 FT. BEFORE I COULD ANSWER; HE CALLED AGAIN; SOMEWHAT AGITATED; AS I HAD NOT ANSWERED QUICK ENOUGH. HE GAVE ME THE DSCNT AND HE GAVE A FREQ CHANGE AT THE SAME TIME. I STARTED DOWN TO 5000 FT AND ATTEMPTED CONTACT ON NEW FREQ. FREQ WAS CONGESTED AND I COULD NOT GET THROUGH VERY SOON. I RECEIVED A TA WHICH TURNED TO AN RA AND I LEVELED OFF AT PRESENT ALT. WARNING STOPPED; THEN BEGAN AGAIN WITH 'CLB.' I WAS AT ABOUT 6500 FT AND CLBED TO 7000 FT. GOT THROUGH TO CTLR AND HE SAID 'WHO ARE YOU; AND WHERE ARE YOU?' MY FO RESPONDED; 11000 FT FOR 5000 FT; 15 NM NE OF DFW. CTLR RESPONDED 'MAINTAIN 7000 FT; YOU ARE DSNDING THROUGH MY DEP CORRIDOR.' HE THEN SAID THE PREVIOUS CTLR HAD DSNDED US THROUGH HIS AIRSPACE AND SAID 'OH MY GOD.' VECTORED TO ARPT FOR UNEVENTFUL LNDG. GND CTLR WHEN QUESTIONED SAID SHE KNEW OF NO EVENT OR PHONE NUMBER FOR US TO CALL. CTLR INFLT WHO CAUGHT THE ERROR ALSO ASSURED US OF NO PROB. I REMEMBER IT THIS WAY. POSSIBILITIES FOR ERROR ARE AS FOLLOWS: 1) CTLR GAVE US WRONG CLRNC AS ARPT HAD JUST CHANGED TO N FLOW FROM S. 2) I TOOK THE WRONG CALL AND IT WAS NOT CAUGHT; AND ACTED ON IT. IT WAS MY FIRST TRIP TO DFW EVER -- AND I WAS DOING IOE (INITIAL TRAINING EXPERIENCE) WITH A BRAND NEW FO; WHO WAS VERY SLOW. I BELIEVE THAT THE CTLR MISTOOK ME FOR ANOTHER FLT; AND TCASII SAVED THE DAY. I MAY BE WRONG BUT I DON'T THINK SO.