theorem Th47: for A,B being functional set holds
SymbolsOf (A\/B) = SymbolsOf A \/ (SymbolsOf B)
proof
let A, B be functional set;
set AF={rng x where x is Element of A: x in A}, BF=
{rng x where x is Element of B: x in B}, F=
{rng x where x is Element of A\/B: x in A\/B};
A null B c= A\/B & B null A c= A\/B; then reconsider
AFF=AF, BFF=BF as Subset of F by Lm51;
A1: AFF \/ BFF c= F;
now
let y be object; assume y in F\BF; then
A2: y in F & not y in BF by XBOOLE_0:def 5; then
consider x being Element of A\/B such that
A3: y=rng x & x in A\/B;
not x in B by A3, A2;
then
A4: x in A null {{}} by A3, XBOOLE_0:def 3; then reconsider xx=x as
Element of A\/{{}};
thus y in AF by A4, A3;
end; then
F\BF \/ BF c= AF \/ BF by XBOOLE_1:9, TARSKI:def 3;
then F null BFF c= AF \/ BF by XBOOLE_1:39; then
A5: AF \/ BF = F by A1;
thus thesis by A5, ZFMISC_1:78;
end;
