reserve X,Y for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,h for Function;

theorem Th7:
  dom f = {x} implies f = {[x,f.x]}
proof
  reconsider g = {[x,f.x]} as Function;
  assume
A1: dom f = {x};
  for y,z being object holds [y,z] in f iff [y,z] in g
  proof let y,z be object;
    thus [y,z] in f implies [y,z] in g
    proof
      assume
A2:   [y,z] in f;
      then y in {x} by A1,XTUPLE_0:def 12;
      then
A3:   y = x by TARSKI:def 1;
A4:   rng f = {f.x} by A1,FUNCT_1:4;
      z in rng f by A2,XTUPLE_0:def 13;
      then z = f.x by A4,TARSKI:def 1;
      hence thesis by A3,TARSKI:def 1;
    end;
    assume [y,z] in g;
    then
A5: y = x & z = f.x by Lm2;
    x in dom f by A1,TARSKI:def 1;
    hence thesis by A5,FUNCT_1:def 2;
  end;
  hence thesis;
end;
