reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

theorem Th6:
  F = id SD iff dom F = SD & for d st d in SD holds F/.d = d
proof
  thus F = id SD implies dom F = SD & for d st d in SD holds F/.d = d
  proof
    assume
A1: F = id SD;
    hence
A2: dom F = SD by RELAT_1:45;
    let d;
    assume
A3: d in SD;
    then (F qua Function).d = d by A1,FUNCT_1:18;
    hence thesis by A2,A3,PARTFUN1:def 6;
  end;
  assume that
A4: dom F = SD and
A5: for d st d in SD holds F/.d = d;
  now
    let x be object;
    assume
A6: x in SD;
    then reconsider x1=x as Element of D;
    F/.x1=x1 by A5,A6;
    hence (F qua Function).x = x by A4,A6,PARTFUN1:def 6;
  end;
  hence thesis by A4,FUNCT_1:17;
end;
