
theorem
  for L being RelStr, S being Subset of L holds id S is Function of
subrelstr S, L & for f being Function of subrelstr S, L st f = id S holds f is
  monotone
proof
  let L be RelStr, S be Subset of L;
A1: the carrier of subrelstr S = S by YELLOW_0:def 15;
A2: rng id S = S by RELAT_1:45;
  dom id S = S by RELAT_1:45;
  hence id S is Function of subrelstr S, L by A1,A2,FUNCT_2:2;
  let f be Function of subrelstr S, L;
  assume
A3: f = id S;
  let x,y be Element of subrelstr S;
  assume
A4: [x,y] in the InternalRel of subrelstr S;
  let a,b be Element of L;
  assume that
A5: a = f.x and
A6: b = f.y;
  x in S by A1,A4,ZFMISC_1:87;
  then
A7: a = x by A3,A5,FUNCT_1:17;
  y in S by A1,A4,ZFMISC_1:87;
  then
A8: b = y by A3,A6,FUNCT_1:17;
  the InternalRel of subrelstr S c= the InternalRel of L by YELLOW_0:def 13;
  hence [a,b] in the InternalRel of L by A4,A7,A8;
end;
