let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN holds B_SUP a,(%I Y) = a
let a be Element of Funcs Y,BOOLEAN ; :: thesis: B_SUP a,(%I Y) = a
A1: for y being Element of Y holds (B_SUP a,(%I Y)) . y = a . y

proof

let y be Element of Y; :: thesis: (B_SUP a,(%I Y)) . y = a . y
A2: y in EqClass y,(%I Y) by EQREL_1:def 8;

A3: now

assume A4: ( ex x being Element of Y st
( x in EqClass y,(%I Y) & a . x = TRUE ) & a . y <> TRUE ) ; :: thesis: contradiction
then consider x1 being Element of Y such that
A5: ( x1 in EqClass y,(%I Y) & a . x1 = TRUE ) ;
( y in EqClass y,(%I Y) & EqClass y,(%I Y) in %I Y ) by EQREL_1:def 8;
then EqClass y,(%I Y) in { B where B is Subset of Y : ex z being set st
( B = {z} & z in Y ) } by PARTIT1:35;
then consider B being Subset of Y such that
A6: ( EqClass y,(%I Y) = B & ex z being set st
( B = {z} & z in Y ) ) ;
consider z being set such that
A7: ( EqClass y,(%I Y) = {z} & z in Y ) by A6;
A8: ( y in {z} & z in Y ) by A7, EQREL_1:def 8;
x1 = z by A5, A7, TARSKI:def 1;
hence contradiction by A4, A5, A8, TARSKI:def 1; :: thesis: verum

end;

now

assume A9: ( ( for x being Element of Y holds
( not x in EqClass y,(%I Y) or not a . x = TRUE ) ) & a . y <> TRUE ) ; :: thesis: (B_SUP a,(%I Y)) . y = a . y
then a . y = FALSE by XBOOLEAN:def 3;
hence (B_SUP a,(%I Y)) . y = a . y by A9, Def20; :: thesis: verum

end;

hence (B_SUP a,(%I Y)) . y = a . y by A2, A3, Def20; :: thesis: verum

end;

consider k3 being Function such that
A10: ( B_SUP a,(%I Y) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A11: ( a = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
for u being set st u in Y holds
k3 . u = k4 . u by A1, A10, A11;
hence B_SUP a,(%I Y) = a by A10, A11, FUNCT_1:9; :: thesis: verum