let x be object ; :: thesis: for X being set
for B being SetSequence of X holds
( x in meet (rng B) iff for n being Nat holds x in B . n )
let X be set ; :: thesis: for B being SetSequence of X holds
( x in meet (rng B) iff for n being Nat holds x in B . n )
let B be SetSequence of X; :: thesis: ( x in meet (rng B) iff for n being Nat holds x in B . n )
A1: dom B = NAT by FUNCT_2:def 1;

A2: now :: thesis: for x being object st ( for n being Nat holds x in B . n ) holds
x in meet (rng B)

let x be object ; :: thesis: ( ( for n being Nat holds x in B . n ) implies x in meet (rng B) )
assume A3: for n being Nat holds x in B . n ; :: thesis: x in meet (rng B)

now :: thesis: for Y being set st Y in rng B holds
x in Y

let Y be set ; :: thesis: ( Y in rng B implies x in Y )
assume Y in rng B ; :: thesis: x in Y
then consider n being object such that
A4: n in NAT and
A5: Y = B . n by A1, FUNCT_1:def 3;
thus x in Y by A3, A4, A5; :: thesis: verum

end;

hence x in meet (rng B) by SETFAM_1:def 1; :: thesis: verum

end;

now :: thesis: for x being object st x in meet (rng B) holds
for n being Nat holds x in B . n

let x be object ; :: thesis: ( x in meet (rng B) implies for n being Nat holds x in B . n )
assume A6: x in meet (rng B) ; :: thesis: for n being Nat holds x in B . n

now :: thesis: for k being Nat holds x in B . k

let k be Nat; :: thesis: x in B . k
k in NAT by ORDINAL1:def 12;
then B . k in rng B by FUNCT_2:4;
hence x in B . k by A6, SETFAM_1:def 1; :: thesis: verum

end;

hence for n being Nat holds x in B . n ; :: thesis: verum

end;

hence ( x in meet (rng B) iff for n being Nat holds x in B . n ) by A2; :: thesis: verum