defpred S_{1}[ object ] means ex B being MSSubset of U0 st

( B in SubSort A & $1 = B . s );

set C = bool (Union the Sorts of U0);

consider X being set such that

A1: for x being object holds

( x in X iff ( x in bool (Union the Sorts of U0) & S_{1}[x] ) )
from XBOOLE_0:sch 1();

take X ; :: thesis: for x being object holds

( x in X iff ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) )

A2: bool (Union the Sorts of U0) = bool (union (rng the Sorts of U0)) by CARD_3:def 4;

for x being set holds

( x in X iff ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) )

( x in X iff ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) ) ; :: thesis: verum

( B in SubSort A & $1 = B . s );

set C = bool (Union the Sorts of U0);

consider X being set such that

A1: for x being object holds

( x in X iff ( x in bool (Union the Sorts of U0) & S

take X ; :: thesis: for x being object holds

( x in X iff ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) )

A2: bool (Union the Sorts of U0) = bool (union (rng the Sorts of U0)) by CARD_3:def 4;

for x being set holds

( x in X iff ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) )

proof

hence
for x being object holds
dom the Sorts of U0 = the carrier of S
by PARTFUN1:def 2;

then the Sorts of U0 . s in rng the Sorts of U0 by FUNCT_1:def 3;

then A3: the Sorts of U0 . s c= union (rng the Sorts of U0) by ZFMISC_1:74;

let x be set ; :: thesis: ( x in X iff ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) )

thus ( x in X implies ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) ) by A1; :: thesis: ( ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) implies x in X )

given B being MSSubset of U0 such that A4: B in SubSort A and

A5: x = B . s ; :: thesis: x in X

B c= the Sorts of U0 by PBOOLE:def 18;

then B . s c= the Sorts of U0 . s ;

then x c= union (rng the Sorts of U0) by A5, A3;

hence x in X by A1, A2, A4, A5; :: thesis: verum

end;then the Sorts of U0 . s in rng the Sorts of U0 by FUNCT_1:def 3;

then A3: the Sorts of U0 . s c= union (rng the Sorts of U0) by ZFMISC_1:74;

let x be set ; :: thesis: ( x in X iff ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) )

thus ( x in X implies ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) ) by A1; :: thesis: ( ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) implies x in X )

given B being MSSubset of U0 such that A4: B in SubSort A and

A5: x = B . s ; :: thesis: x in X

B c= the Sorts of U0 by PBOOLE:def 18;

then B . s c= the Sorts of U0 . s ;

then x c= union (rng the Sorts of U0) by A5, A3;

hence x in X by A1, A2, A4, A5; :: thesis: verum

( x in X iff ex B being MSSubset of U0 st

( B in SubSort A & x = B . s ) ) ; :: thesis: verum