Step |
Hyp |
Ref |
Expression |
1 |
|
coe1fval.a |
|- A = ( coe1 ` F ) |
2 |
1
|
coe1fval |
|- ( F e. V -> A = ( n e. NN0 |-> ( F ` ( 1o X. { n } ) ) ) ) |
3 |
2
|
fveq1d |
|- ( F e. V -> ( A ` N ) = ( ( n e. NN0 |-> ( F ` ( 1o X. { n } ) ) ) ` N ) ) |
4 |
|
sneq |
|- ( n = N -> { n } = { N } ) |
5 |
4
|
xpeq2d |
|- ( n = N -> ( 1o X. { n } ) = ( 1o X. { N } ) ) |
6 |
5
|
fveq2d |
|- ( n = N -> ( F ` ( 1o X. { n } ) ) = ( F ` ( 1o X. { N } ) ) ) |
7 |
|
eqid |
|- ( n e. NN0 |-> ( F ` ( 1o X. { n } ) ) ) = ( n e. NN0 |-> ( F ` ( 1o X. { n } ) ) ) |
8 |
|
fvex |
|- ( F ` ( 1o X. { N } ) ) e. _V |
9 |
6 7 8
|
fvmpt |
|- ( N e. NN0 -> ( ( n e. NN0 |-> ( F ` ( 1o X. { n } ) ) ) ` N ) = ( F ` ( 1o X. { N } ) ) ) |
10 |
3 9
|
sylan9eq |
|- ( ( F e. V /\ N e. NN0 ) -> ( A ` N ) = ( F ` ( 1o X. { N } ) ) ) |