
theorem Th13:
  for A,B being set holds B c= A implies (id A).:(B) = B
proof
  let A,B be set;
  assume
A1: B c= A;
  thus (id A).:(B) c= B
  proof
    let y be object;
    assume y in (id A).:(B);
    then ex x being object st x in dom(id A) & x in B & (id A).x = y by
FUNCT_1:def 6;
    hence thesis by FUNCT_1:17;
  end;
  let y be object;
  assume
A2: y in B;
  then dom(id A) = A & (id A).y = y by A1,FUNCT_1:17;
  hence thesis by A1,A2,FUNCT_1:def 6;
end;
