defpred S₁[ object , object ] means ex H being strict Subgroup of G st
( $2 = H & $1 = the carrier of H );
defpred S₂[ set ] means ex H being strict Subgroup of G st $1 = the carrier of H;
consider B being set such that
A1: for x being set holds
( x in B iff ( x in bool the carrier of G & S₂[x] ) ) from XFAMILY:sch 1();
A2: for x, y1, y2 being object st S₁[x,y1] & S₁[x,y2] holds
y1 = y2 by Th59;
consider f being Function such that
A3: for x, y being object holds
( [x,y] in f iff ( x in B & S₁[x,y] ) ) from FUNCT_1:sch 1(A2);
for x being object holds
( x in B iff ex y being object st [x,y] in f )

proof

let x be object ; :: thesis: ( x in B iff ex y being object st [x,y] in f )
thus ( x in B implies ex y being object st [x,y] in f ) :: thesis: ( ex y being object st [x,y] in f implies x in B )

proof

assume A4: x in B ; :: thesis: ex y being object st [x,y] in f
then consider H being strict Subgroup of G such that
A5: x = the carrier of H by A1;
take H ; :: thesis: [x,H] in f
thus [x,H] in f by A3, A4, A5; :: thesis: verum

end;

thus ( ex y being object st [x,y] in f implies x in B ) by A3; :: thesis: verum

end;

then A6: B = dom f by XTUPLE_0:def 12;
for y being object holds
( y in rng f iff y is strict Subgroup of G )

proof

let y be object ; :: thesis: ( y in rng f iff y is strict Subgroup of G )
thus ( y in rng f implies y is strict Subgroup of G ) :: thesis: ( y is strict Subgroup of G implies y in rng f )

proof

assume y in rng f ; :: thesis: y is strict Subgroup of G
then consider x being object such that
A7: ( x in dom f & y = f . x ) by FUNCT_1:def 3;
[x,y] in f by A7, FUNCT_1:def 2;
then ex H being strict Subgroup of G st
( y = H & x = the carrier of H ) by A3;
hence y is strict Subgroup of G ; :: thesis: verum

end;

assume y is strict Subgroup of G ; :: thesis: y in rng f
then reconsider H = y as strict Subgroup of G ;
A8: y is set by TARSKI:1;
reconsider x = the carrier of H as set ;
A8a: the carrier of H c= the carrier of G by DefA5;
then A9: x in dom f by A1, A6;
[x,y] in f by A1, A3, A8a;
then y = f . x by A9, FUNCT_1:def 2, A8;
hence y in rng f by A9, FUNCT_1:def 3; :: thesis: verum

end;

hence ex b₁ being set st
for x being object holds
( x in b₁ iff x is strict Subgroup of G ) ; :: thesis: verum