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;
reserve f for BinOp of D;
reserve a,a1,a2,b,b1,b2,A,B,C,X,Y,Z,x,x1,x2,y,y1,y2,z for set,
U,U1,U2,U3 for non empty set, u,u1,u2 for Element of U,
P,Q,R for Relation, f,f1,f2,g,g1,g2 for Function,
k,m,n for Nat, kk,mm,nn for Element of NAT, m1, n1 for non zero Nat,
p, p1, p2 for FinSequence, q, q1, q2 for U-valued FinSequence;

theorem Th45: SymbolsOf {f} = rng f ::#Th45
proof
set P=f, X={P}, F={rng x where x is Element of X\/{{}}: x in X}, LH=union F,
RH=rng P;
X null {{}} c= X\/{{}}; then reconsider XX=X as Subset of X\/{{}};
reconsider PP=P as Element of XX by TARSKI:def 1;
reconsider PPP=PP as Element of X\/{{}} by TARSKI:def 3;
now
let y be object; assume y in LH; then consider z such that
A1: y in z & z in F by TARSKI:def 4; consider x being Element of X\/{{}}
such that
A2: z=rng x & x in X by A1;
thus y in RH by A1, A2, TARSKI:def 1;
end; then
A3: LH c= RH;
now
let y be object; assume y in RH; then y in rng PP & rng PPP in F; hence
y in LH by TARSKI:def 4;
end; then
RH c= LH; hence thesis by A3;
end;
