let X1, X2, X3, X4, X5, X6, X7, Z be set ; :: thesis: ( ( for y being set holds
( y in Z iff ex x1, x2, x3, x4, x5, x6, x7 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & y = [x1,x2,x3,x4,x5,x6,x7] ) ) ) implies Z = [:X1,X2,X3,X4,X5,X6,X7:] )
assume A1: for y being set holds
( y in Z iff ex x1, x2, x3, x4, x5, x6, x7 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & y = [x1,x2,x3,x4,x5,x6,x7] ) ) ; :: thesis: Z = [:X1,X2,X3,X4,X5,X6,X7:]

now

let y be set ; :: thesis: ( ( y in Z implies y in [:[:X1,X2,X3,X4,X5,X6:],X7:] ) & ( y in [:[:X1,X2,X3,X4,X5,X6:],X7:] implies y in Z ) )
thus ( y in Z implies y in [:[:X1,X2,X3,X4,X5,X6:],X7:] ) :: thesis: ( y in [:[:X1,X2,X3,X4,X5,X6:],X7:] implies y in Z )

proof

assume y in Z ; :: thesis: y in [:[:X1,X2,X3,X4,X5,X6:],X7:]
then consider x1, x2, x3, x4, x5, x6, x7 being set such that
A2: ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & y = [x1,x2,x3,x4,x5,x6,x7] ) by A1;
( [x1,x2,x3,x4,x5,x6] in [:X1,X2,X3,X4,X5,X6:] & x7 in X7 ) by A2, Th32;
hence y in [:[:X1,X2,X3,X4,X5,X6:],X7:] by A2, ZFMISC_1:def 2; :: thesis: verum

end;

assume y in [:[:X1,X2,X3,X4,X5,X6:],X7:] ; :: thesis: y in Z
then consider x123456, x7 being set such that
A3: x123456 in [:X1,X2,X3,X4,X5,X6:] and
A4: x7 in X7 and
A5: y = [x123456,x7] by ZFMISC_1:def 2;
consider x1, x2, x3, x4, x5, x6 being set such that
A6: ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x123456 = [x1,x2,x3,x4,x5,x6] ) by A3, Th31;
y = [x1,x2,x3,x4,x5,x6,x7] by A5, A6;
hence y in Z by A1, A4, A6; :: thesis: verum

end;

hence Z = [:X1,X2,X3,X4,X5,X6,X7:] by TARSKI:2; :: thesis: verum