theorem for I being Element of U-InterpretersOf S st l is X-absent &
X is I-satisfied holds X is (l,u) ReassignIn I-satisfied
proof
set II=U-InterpretersOf S; let I be Element of II; set O=OwnSymbolsOf S,
I2=(l,u) ReassignIn I, f2=l.-->({}.-->u); assume
A1: l is X-absent & X is I-satisfied;
now
let phi; reconsider no=rng phi/\O as Subset of rng phi; assume
A2: phi in X; then reconsider Phi={phi} as Subset of X by ZFMISC_1:31;
A3: I-TruthEval phi=1 by A1, A2;
l is (X/\Phi)-absent by A1; then not l in no by FOMODEL2:28; then
{l} misses no by ZFMISC_1:50; then dom f2 misses no; then
I|no +* (f2|no) = I|no null {} by RELAT_1:66; then
I2|no=I|no by FUNCT_4:71; hence I2-TruthEval phi=1 by A3, Th13;
end; hence thesis;
end;
