Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
2 |
|
simp2 |
|- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> G : I --> NN0 ) |
3 |
|
simp1 |
|- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> F e. D ) |
4 |
|
id |
|- ( F e. D -> F e. D ) |
5 |
1
|
psrbagf |
|- ( F e. D -> F : I --> NN0 ) |
6 |
5
|
ffnd |
|- ( F e. D -> F Fn I ) |
7 |
4 6
|
fndmexd |
|- ( F e. D -> I e. _V ) |
8 |
7
|
3ad2ant1 |
|- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> I e. _V ) |
9 |
1
|
psrbag |
|- ( I e. _V -> ( F e. D <-> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) ) |
10 |
8 9
|
syl |
|- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( F e. D <-> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) ) |
11 |
3 10
|
mpbid |
|- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) |
12 |
11
|
simprd |
|- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( `' F " NN ) e. Fin ) |
13 |
1
|
psrbaglesupp |
|- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( `' G " NN ) C_ ( `' F " NN ) ) |
14 |
12 13
|
ssfid |
|- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( `' G " NN ) e. Fin ) |
15 |
1
|
psrbag |
|- ( I e. _V -> ( G e. D <-> ( G : I --> NN0 /\ ( `' G " NN ) e. Fin ) ) ) |
16 |
8 15
|
syl |
|- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( G e. D <-> ( G : I --> NN0 /\ ( `' G " NN ) e. Fin ) ) ) |
17 |
2 14 16
|
mpbir2and |
|- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> G e. D ) |