reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;
reserve X for non empty set,
  Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function
  of Y,X,
  x,x1,x2 for Element of X;
reserve y for Element of Y;
reserve Y for non empty set,
  F for BinOp of X,
  f for Function of Y,X,
  x for Element of X,
  y for Element of Y;
reserve a,b,c for set;
reserve x,y,z for object;

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;
