reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

theorem
  f is one-to-one implies (d in rng f & c = (f").d iff c in dom f & d = f.c)
proof
A1: f" = (f qua Function)";
  assume f is one-to-one;
  hence thesis by A1,FUNCT_1:32;
end;
