DURING ENRTE DEP PHASE FROM DFW; E120 FLT CREW IS CHALLENGED BY D10 CTLR ABOUT ASSIGNED ALT. CLRNC NOT RESOLVED. ATC THOUGHT A HIGHER ALT WAS ISSUED; DIFFERING FROM THAT WHICH WAS ACKNOWLEDGED AND FLOWN BY FLT CREW.

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I AM THE FO ON THE E120. THE CAPT AND I WERE BEGINNING OUR SECOND ROUND TRIP OUT OF DALLAS-FORT WORTH. WE HAD DEPARTED DFW; I WAS THE PNF AND TALKING ON THE RADIOS WHILE THE CAPT WAS FLYING. WE WERE CLRED TO 2000 FT VIA OUR SID; HAD TALKED TO DEP; WHERE HE CLRED US TO 4000 FT. ONCE AT 4000 FT; ATC ASKED US IF WE WANTED TO STAY AT 4000 FT OR EXPEDITE TO 5000 FT. AFTER CONFERRING WITH THE CAPT; I STATED WE WOULD EXPEDITE TO 5000 FT. AFTER LEVELING OUT AT 5000 FT; ATC QUESTIONED 'DO YOU SEE THE ACFT AT YOUR 3 O'CLOCK POS? EXPEDITE THROUGH 6000 FT.' WE BEGAN OUR CLB AND THE CAPT KEYED THE RADIO AND QUERIED WHAT ALT WE WERE CLRED. THIS IS THE FIRST TIME WE HEARD THAT ATC HAD WANTED US TO CLB TO 12000 FT. ATC SEEMED RUSHED AND ASKED THE CAPT WHAT ALT HE HAD CLRED US TO. THE CAPT STATED 4000 FT; AND ATC REPLIED THAT WE WERE AT 5000 FT. WE LEFT IT AT THAT AND CONTINUED OUR CLB TO 12000 FT. THERE WERE MANY OTHER ACFT THAT DEP WAS WORKING WITH THAT AFTERNOON AND HE DID NOT CLARIFY HIS CLRNC TO US. WE WERE FORTUNATE THAT THE CREW WAS ON THEIR FIRST DAY; THE CAPT WAS WELL RESTED AND HAD GOOD PERCEPTION TO QUESTION ATC WHEN THINGS DID NOT SEEM CORRECT.