let Y be non empty TopSpace; :: thesis: for A being Subset of Y holds MaxADSet A c= meet { G where G is Subset of Y : ( G is open & A c= G ) }

let A be Subset of Y; :: thesis: MaxADSet A c= meet { G where G is Subset of Y : ( G is open & A c= G ) }

set F = { G where G is Subset of Y : ( G is open & A c= G ) } ;

[#] Y in { G where G is Subset of Y : ( G is open & A c= G ) } ;

then A1: { G where G is Subset of Y : ( G is open & A c= G ) } <> {} ;

{ G where G is Subset of Y : ( G is open & A c= G ) } c= bool the carrier of Y

let A be Subset of Y; :: thesis: MaxADSet A c= meet { G where G is Subset of Y : ( G is open & A c= G ) }

set F = { G where G is Subset of Y : ( G is open & A c= G ) } ;

[#] Y in { G where G is Subset of Y : ( G is open & A c= G ) } ;

then A1: { G where G is Subset of Y : ( G is open & A c= G ) } <> {} ;

{ G where G is Subset of Y : ( G is open & A c= G ) } c= bool the carrier of Y

proof

then reconsider F = { G where G is Subset of Y : ( G is open & A c= G ) } as Subset-Family of Y ;
let C be object ; :: according to TARSKI:def 3 :: thesis: ( not C in { G where G is Subset of Y : ( G is open & A c= G ) } or C in bool the carrier of Y )

assume C in { G where G is Subset of Y : ( G is open & A c= G ) } ; :: thesis: C in bool the carrier of Y

then ex P being Subset of Y st

( C = P & P is open & A c= P ) ;

hence C in bool the carrier of Y ; :: thesis: verum

end;assume C in { G where G is Subset of Y : ( G is open & A c= G ) } ; :: thesis: C in bool the carrier of Y

then ex P being Subset of Y st

( C = P & P is open & A c= P ) ;

hence C in bool the carrier of Y ; :: thesis: verum

now :: thesis: for C being set st C in F holds

MaxADSet A c= C

hence
MaxADSet A c= meet { G where G is Subset of Y : ( G is open & A c= G ) }
by A1, SETFAM_1:5; :: thesis: verumMaxADSet A c= C

let C be set ; :: thesis: ( C in F implies MaxADSet A c= C )

assume C in F ; :: thesis: MaxADSet A c= C

then ex G being Subset of Y st

( G = C & G is open & A c= G ) ;

hence MaxADSet A c= C by Th38; :: thesis: verum

end;assume C in F ; :: thesis: MaxADSet A c= C

then ex G being Subset of Y st

( G = C & G is open & A c= G ) ;

hence MaxADSet A c= C by Th38; :: thesis: verum