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;

theorem
  for F being Function st X c= F"X1 holds F.:X c= X1
proof
  let F be Function;
  assume X c= F"X1;
  then
A1: F.:X c= F.:F"X1 by RELAT_1:123;
  F.:F"X1 c= X1 by FUNCT_1:75;
  hence thesis by A1;
end;
