Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
2 |
|
id |
|- ( F e. D -> F e. D ) |
3 |
1
|
psrbagf |
|- ( F e. D -> F : I --> NN0 ) |
4 |
3
|
ffnd |
|- ( F e. D -> F Fn I ) |
5 |
2 4
|
fndmexd |
|- ( F e. D -> I e. _V ) |
6 |
1
|
psrbag |
|- ( I e. _V -> ( F e. D <-> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) ) |
7 |
6
|
biimpa |
|- ( ( I e. _V /\ F e. D ) -> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) |
8 |
5 7
|
mpancom |
|- ( F e. D -> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) |
9 |
8
|
simprd |
|- ( F e. D -> ( `' F " NN ) e. Fin ) |
10 |
|
frnnn0fsuppg |
|- ( ( F e. D /\ F : I --> NN0 ) -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) ) |
11 |
3 10
|
mpdan |
|- ( F e. D -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) ) |
12 |
9 11
|
mpbird |
|- ( F e. D -> F finSupp 0 ) |