reserve A,B,a,b,c,d,e,f,g,h for set;

theorem Th1:
  (id A)|B = id A /\ [:B,B:]
proof
  thus (id A)|B c= id A /\ [:B,B:]
  proof
    let a be object;
    assume
A1: a in (id A)|B;
    (id A)|B is Relation of B,A by RELSET_1:18;
    then consider x,y being object such that
A2: a = [x,y] and
A3: x in B and
    y in A by A1,RELSET_1:2;
A4: [x,y] in id A by A1,A2,RELAT_1:def 11;
    then x = y by RELAT_1:def 10;
    then [x,y] in [:B,B:] by A3,ZFMISC_1:87;
    hence thesis by A2,A4,XBOOLE_0:def 4;
  end;
  let a be object;
  assume
A5: a in id A /\ [:B,B:];
  then a in [:B,B:] by XBOOLE_0:def 4;
  then
A6: ex x1,y1 being object st x1 in B & y1 in B & a = [x1,y1]
by ZFMISC_1:def 2;
  a in id A by A5,XBOOLE_0:def 4;
  hence thesis by A6,RELAT_1:def 11;
end;
