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;
