defpred S₁[ set ] means ex E being non empty finite Subset of I ex f being SigmaSection of E,F st
( E c= J & $1 = meet (rng f) );
consider X being set such that
A1: for x being set holds
( x in X iff ( x in bool Omega & S₁[x] ) ) from XFAMILY:sch 1();
for x being object st x in X holds
x in bool Omega by A1;
then reconsider X = X as Subset-Family of Omega by TARSKI:def 3;
take X ; :: thesis: for x being Subset of Omega holds
( x in X iff ex E being non empty finite Subset of I ex f being SigmaSection of E,F st
( E c= J & x = meet (rng f) ) )
let x be Subset of Omega; :: thesis: ( x in X iff ex E being non empty finite Subset of I ex f being SigmaSection of E,F st
( E c= J & x = meet (rng f) ) )
thus ( x in X iff ex E being non empty finite Subset of I ex f being SigmaSection of E,F st
( E c= J & x = meet (rng f) ) ) by A1; :: thesis: verum