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Metamath Proof Explorer

Theorem satfvsuclem1

Description: Lemma 1 for satfvsuc . (Contributed by AV, 8-Oct-2023)

Ref Expression
Hypothesis satfv0.s 
|- S = ( M Sat E )
Assertion satfvsuclem1 
|- ( ( 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 )

Proof

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 )