Step |
Hyp |
Ref |
Expression |
1 |
|
satfv0fun |
|- ( ( M e. V /\ E e. W ) -> Fun ( ( M Sat E ) ` (/) ) ) |
2 |
|
satffunlem1lem1 |
|- ( Fun ( ( M Sat E ) ` (/) ) -> Fun { <. x , y >. | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } ) |
3 |
1 2
|
syl |
|- ( ( M e. V /\ E e. W ) -> Fun { <. x , y >. | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } ) |
4 |
|
satffunlem1lem2 |
|- ( ( M e. V /\ E e. W ) -> ( dom ( ( M Sat E ) ` (/) ) i^i dom { <. x , y >. | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } ) = (/) ) |
5 |
|
funun |
|- ( ( ( Fun ( ( M Sat E ) ` (/) ) /\ Fun { <. x , y >. | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } ) /\ ( dom ( ( M Sat E ) ` (/) ) i^i dom { <. x , y >. | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } ) = (/) ) -> Fun ( ( ( M Sat E ) ` (/) ) u. { <. x , y >. | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } ) ) |
6 |
1 3 4 5
|
syl21anc |
|- ( ( M e. V /\ E e. W ) -> Fun ( ( ( M Sat E ) ` (/) ) u. { <. x , y >. | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } ) ) |
7 |
|
peano1 |
|- (/) e. _om |
8 |
|
eqid |
|- ( M Sat E ) = ( M Sat E ) |
9 |
8
|
satfvsuc |
|- ( ( M e. V /\ E e. W /\ (/) e. _om ) -> ( ( M Sat E ) ` suc (/) ) = ( ( ( M Sat E ) ` (/) ) u. { <. x , y >. | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } ) ) |
10 |
7 9
|
mp3an3 |
|- ( ( M e. V /\ E e. W ) -> ( ( M Sat E ) ` suc (/) ) = ( ( ( M Sat E ) ` (/) ) u. { <. x , y >. | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } ) ) |
11 |
10
|
funeqd |
|- ( ( M e. V /\ E e. W ) -> ( Fun ( ( M Sat E ) ` suc (/) ) <-> Fun ( ( ( M Sat E ) ` (/) ) u. { <. x , y >. | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } ) ) ) |
12 |
6 11
|
mpbird |
|- ( ( M e. V /\ E e. W ) -> Fun ( ( M Sat E ) ` suc (/) ) ) |