reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;
reserve e,u,v for object, E,X,Y,X1 for set;
reserve X,Y,Z for non empty set;

theorem
  for F being Function of X,Y, y being Element of Y,
      z being Element of Z holds [:F,id Z:]"{[y,z]} = [:F"{y},{z}:]
proof
  let F be Function of X,Y, y be Element of Y, z be Element of Z;
  thus [:F,id Z:]"{[y,z]} = [:F,id Z:]"[:{y},{z}:] by ZFMISC_1:29
    .= [:F"{y},(id Z)"{z}:] by FUNCT_3:73
    .= [:F"{y},{z}:] by FUNCT_2:94;
end;
