Appearing a unique number of times
05AB1E, 6 bytes

D¢DD¢Q


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Outputs 1 if each item appears a unique number of times, 0 otherwise.

D           # duplicate the input
 ¢          # count occurences of each item in the input
  DD        # triplicate the list of occurence counts
    ¢       # count occurences of each occurence count in the list of occurence counts
     Q      # check whether this is equal to the list of occurence counts


Without the source restriction, this could be 4 bytes: ¢Ð¢Q.