reserve x, x1, x2, y, z, X9 for set,
  X, Y for finite set,
  n, k, m for Nat,
  f for Function;
reserve F,Ch for Function;

theorem
  union rng F is non empty & Intersection(F,Ch,y)=union rng F implies F|
  Ch"{y}=(Ch"{y}-->union rng F)
proof
  set ChF=Ch"{y}-->union rng F;
  assume that
A1: union rng F is non empty and
A2: Intersection(F,Ch,y) = union rng F;
A3: dom F/\Ch"{y} =dom (F|Ch"{y}) by RELAT_1:61;
  dom F/\Ch"{y}=Ch"{y} by A1,A2,Th19,XBOOLE_1:28;
  then
A4: dom (F|Ch"{y})=dom ChF by A3;
  assume F|Ch"{y}<>ChF;
  then consider x being object such that
A5: x in dom (F|Ch"{y}) and
A6: (F|Ch"{y}).x<>ChF.x by A4;
  x in dom F /\ Ch"{y} by A5,RELAT_1:61;
  then
A7: x in dom F by XBOOLE_0:def 4;
  x in dom F /\ Ch"{y} by A5,RELAT_1:61;
  then
A8: x in Ch"{y} by XBOOLE_0:def 4;
  then
A9: ChF.x=union rng F by FUNCOP_1:7;
  Ch.x in {y} by A8,FUNCT_1:def 7;
  then
A10: Ch.x =y by TARSKI:def 1;
  F.x=(F|Ch"{y}).x by A5,FUNCT_1:47;
  then (F|Ch"{y}).x in rng F by A7,FUNCT_1:def 3;
  then (F|Ch"{y}).x c= ChF.x by A9,ZFMISC_1:74;
  then not union rng F c= (F|Ch"{y}).x by A6,A9;
  then consider z being object such that
A11: z in union rng F and
A12: not z in (F|Ch"{y}).x;
  x in dom Ch by A8,FUNCT_1:def 7;
  then z in F.x by A2,A11,A10,Def2;
  hence thesis by A5,A12,FUNCT_1:47;
end;
