theorem Th82:
  x .--> y is_isomorphism_of {[x,x]},{[y,y]}
proof
  set F = x .--> y;
  set R = {[x,x]};
  set S = {[y,y]};
A1: field R = {x} by RELAT_1:173;
  hence dom F = field R;
  field S = {y} by RELAT_1:173;
  hence rng F = field S by RELAT_1:160;
  thus F is one-to-one;
  let a,b be object;
  hereby
    assume [a,b] in R;
    then [a,b] = [x,x] by TARSKI:def 1;
    then
A2: a = x & b = x by XTUPLE_0:1;
    hence a in field R & b in field R by A1,TARSKI:def 1;
    F.x = y by Th72;
    hence [F.a,F.b] in S by A2,TARSKI:def 1;
  end;
  assume a in field R & b in field R;
  then a = x & b = x by A1,TARSKI:def 1;
  hence thesis by TARSKI:def 1;
end;
