Step |
Hyp |
Ref |
Expression |
1 |
|
resfifsupp.f |
|- ( ph -> Fun F ) |
2 |
|
resfifsupp.x |
|- ( ph -> X e. Fin ) |
3 |
|
resfifsupp.z |
|- ( ph -> Z e. V ) |
4 |
|
funrel |
|- ( Fun F -> Rel F ) |
5 |
1 4
|
syl |
|- ( ph -> Rel F ) |
6 |
|
resindm |
|- ( Rel F -> ( F |` ( X i^i dom F ) ) = ( F |` X ) ) |
7 |
5 6
|
syl |
|- ( ph -> ( F |` ( X i^i dom F ) ) = ( F |` X ) ) |
8 |
1
|
funfnd |
|- ( ph -> F Fn dom F ) |
9 |
|
fnresin2 |
|- ( F Fn dom F -> ( F |` ( X i^i dom F ) ) Fn ( X i^i dom F ) ) |
10 |
8 9
|
syl |
|- ( ph -> ( F |` ( X i^i dom F ) ) Fn ( X i^i dom F ) ) |
11 |
|
infi |
|- ( X e. Fin -> ( X i^i dom F ) e. Fin ) |
12 |
2 11
|
syl |
|- ( ph -> ( X i^i dom F ) e. Fin ) |
13 |
10 12 3
|
fndmfifsupp |
|- ( ph -> ( F |` ( X i^i dom F ) ) finSupp Z ) |
14 |
7 13
|
eqbrtrrd |
|- ( ph -> ( F |` X ) finSupp Z ) |