Step |
Hyp |
Ref |
Expression |
1 |
|
satfv0.s |
|- S = ( M Sat E ) |
2 |
|
ancom |
|- ( ( E. u e. ( S ` N ) ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) /\ y e. ~P ( M ^m _om ) ) <-> ( y e. ~P ( M ^m _om ) /\ E. u e. ( S ` N ) ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) ) ) |
3 |
2
|
opabbii |
|- { <. x , y >. | ( E. u e. ( S ` N ) ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) /\ y e. ~P ( M ^m _om ) ) } = { <. x , y >. | ( y e. ~P ( M ^m _om ) /\ E. u e. ( S ` N ) ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) ) } |
4 |
|
ovex |
|- ( M ^m _om ) e. _V |
5 |
4
|
pwex |
|- ~P ( M ^m _om ) e. _V |
6 |
5
|
a1i |
|- ( ( M e. V /\ E e. W /\ N e. _om ) -> ~P ( M ^m _om ) e. _V ) |
7 |
|
fvex |
|- ( S ` N ) e. _V |
8 |
|
unab |
|- ( { x | E. v e. ( S ` N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) } u. { x | E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) } ) = { x | ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } |
9 |
7
|
abrexex |
|- { x | E. v e. ( S ` N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) } e. _V |
10 |
|
simpl |
|- ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) -> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) |
11 |
10
|
reximi |
|- ( E. v e. ( S ` N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) -> E. v e. ( S ` N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) |
12 |
11
|
ss2abi |
|- { x | E. v e. ( S ` N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) } C_ { x | E. v e. ( S ` N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) } |
13 |
9 12
|
ssexi |
|- { x | E. v e. ( S ` N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) } e. _V |
14 |
|
omex |
|- _om e. _V |
15 |
14
|
abrexex |
|- { x | E. i e. _om x = A.g i ( 1st ` u ) } e. _V |
16 |
|
simpl |
|- ( ( x = A.g i ( 1st ` u ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) -> x = A.g i ( 1st ` u ) ) |
17 |
16
|
reximi |
|- ( E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) -> E. i e. _om x = A.g i ( 1st ` u ) ) |
18 |
17
|
ss2abi |
|- { x | E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) } C_ { x | E. i e. _om x = A.g i ( 1st ` u ) } |
19 |
15 18
|
ssexi |
|- { x | E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) } e. _V |
20 |
13 19
|
unex |
|- ( { x | E. v e. ( S ` N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) } u. { x | E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) } ) e. _V |
21 |
8 20
|
eqeltrri |
|- { x | ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } e. _V |
22 |
21
|
a1i |
|- ( ( ( ( M e. V /\ E e. W /\ N e. _om ) /\ y e. ~P ( M ^m _om ) ) /\ u e. ( S ` N ) ) -> { x | ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } e. _V ) |
23 |
22
|
ralrimiva |
|- ( ( ( M e. V /\ E e. W /\ N e. _om ) /\ y e. ~P ( M ^m _om ) ) -> A. u e. ( S ` N ) { x | ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } e. _V ) |
24 |
|
abrexex2g |
|- ( ( ( S ` N ) e. _V /\ A. u e. ( S ` N ) { x | ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } e. _V ) -> { x | E. u e. ( S ` N ) ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } e. _V ) |
25 |
7 23 24
|
sylancr |
|- ( ( ( M e. V /\ E e. W /\ N e. _om ) /\ y e. ~P ( M ^m _om ) ) -> { x | E. u e. ( S ` N ) ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) } e. _V ) |
26 |
6 25
|
opabex3rd |
|- ( ( M e. V /\ E e. W /\ N e. _om ) -> { <. x , y >. | ( y e. ~P ( M ^m _om ) /\ E. u e. ( S ` N ) ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) ) } e. _V ) |
27 |
3 26
|
eqeltrid |
|- ( ( M e. V /\ E e. W /\ N e. _om ) -> { <. x , y >. | ( E. u e. ( S ` N ) ( E. v e. ( S ` N ) ( 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 = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) ) /\ y e. ~P ( M ^m _om ) ) } e. _V ) |