reserve A,B,C,Y,x,y,z for set, U, D for non empty set,
X for non empty Subset of D, d,d1,d2 for Element of D;
reserve P,Q,R for Relation, g for Function, p,q for FinSequence;
reserve f for BinOp of D, i,m,n for Nat;
reserve X for set, f for Function;
reserve U1,U2 for non empty set;

theorem for y being non empty set holds (y=f.x iff x in f"{y}) ::#Th25
proof
let y be non empty set; thus y=f.x implies x in f"{y}
proof
assume y=f.x; then x in dom f & f.x in {y} by FUNCT_1:def 2, TARSKI:def 1;
hence thesis by FUNCT_1:def 7;
end;
assume x in f"{y}; then x in dom f & f.x in {y} by FUNCT_1:def 7;
hence thesis by TARSKI:def 1;
end;
