let X1, X2, X3, X4, X5, Z be set ; :: thesis: ( ( for y being set holds
( y in Z iff ex x1, x2, x3, x4, x5 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & y = [x1,x2,x3,x4,x5] ) ) ) implies Z = [:X1,X2,X3,X4,X5:] )
assume A1: for y being set holds
( y in Z iff ex x1, x2, x3, x4, x5 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & y = [x1,x2,x3,x4,x5] ) ) ; :: thesis: Z = [:X1,X2,X3,X4,X5:]

now

let y be set ; :: thesis: ( ( y in Z implies y in [:[:X1,X2,X3,X4:],X5:] ) & ( y in [:[:X1,X2,X3,X4:],X5:] implies y in Z ) )
thus ( y in Z implies y in [:[:X1,X2,X3,X4:],X5:] ) :: thesis: ( y in [:[:X1,X2,X3,X4:],X5:] implies y in Z )

proof

assume y in Z ; :: thesis: y in [:[:X1,X2,X3,X4:],X5:]
then consider x1, x2, x3, x4, x5 being set such that
A2: ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 ) and
A3: ( x5 in X5 & y = [x1,x2,x3,x4,x5] ) by A1;
[x1,x2,x3,x4] in [:X1,X2,X3,X4:] by A2, MCART_1:80;
hence y in [:[:X1,X2,X3,X4:],X5:] by A3, ZFMISC_1:def 2; :: thesis: verum

end;

assume y in [:[:X1,X2,X3,X4:],X5:] ; :: thesis: y in Z
then consider x1234, x5 being set such that
A4: x1234 in [:X1,X2,X3,X4:] and
A5: x5 in X5 and
A6: y = [x1234,x5] by ZFMISC_1:def 2;
consider x1, x2, x3, x4 being set such that
A7: ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 ) and
A8: x1234 = [x1,x2,x3,x4] by A4, MCART_1:79;
y = [x1,x2,x3,x4,x5] by A6, A8;
hence y in Z by A1, A5, A7; :: thesis: verum

end;

hence Z = [:X1,X2,X3,X4,X5:] by TARSKI:1; :: thesis: verum