reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  for f being Relation, x, y st dom f = {x} & rng f = {y}
  holds f = { [x,y] }
proof
  let f be Relation, x, y;
A1: f c= [:dom f,rng f:] by Th1;
  assume
A2: dom f = {x} & rng f = {y};
  x in dom f by A2,TARSKI:def 1;
  then consider yy being object such that
A3: [x,yy] in f by XTUPLE_0:def 12;
  yy in rng f by A3,XTUPLE_0:def 13;
  then [x,y] in f by A3,A2,TARSKI:def 1;
 hence thesis by A1,A2,ZFMISC_1:29,31;
end;
