reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;
reserve A,B for set;
reserve x,y,i,j,k for object;
reserve x for set;
reserve x for object;

theorem
  for R being Relation st dom R = {x} & rng R = {y}
    holds R = x .--> y
proof let R be Relation;
  assume that
A1: dom R = {x} and
A2: rng R = {y};
  set g = x .--> y;
A3: g = {[x,y]} by ZFMISC_1:29;
  for a, b being object holds [a,b] in R iff [a,b] in g
  proof
    let a, b be object;
    hereby
      assume
A4:   [a,b] in R;
      then b in rng R by XTUPLE_0:def 13;
      then
A5:   b = y by A2,TARSKI:def 1;
      a in dom R by A4,XTUPLE_0:def 12;
      then a = x by A1,TARSKI:def 1;
      hence [a,b] in g by A3,A5,TARSKI:def 1;
    end;
    assume [a,b] in g;
    then
A6: [a,b] = [x,y] by A3,TARSKI:def 1;
    then
A7: b = y by XTUPLE_0:1;
    a = x by A6,XTUPLE_0:1;
    then a in dom R by A1,TARSKI:def 1;
    then consider z being object such that
A8: [a,z] in R by XTUPLE_0:def 12;
    z in rng R by A8,XTUPLE_0:def 13;
    hence thesis by A2,A7,A8,TARSKI:def 1;
  end;
  hence thesis by RELAT_1:def 2;
end;
