Metamath Proof Explorer

Theorem satffunlem2lem2

Description: Lemma 2 for satffunlem2 . (Contributed by AV, 27-Oct-2023)

Ref Expression
Hypotheses satffunlem2lem2.s 
|- S = ( M Sat E )
satffunlem2lem2.a 
|- A = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) )
satffunlem2lem2.b 
|- B = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) }
Assertion satffunlem2lem2 
|- ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( dom ( S ` suc N ) i^i dom { <. x , y >. | ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) ) } ) = (/) )

Proof

Step Hyp Ref Expression
1 satffunlem2lem2.s 
 |-  S = ( M Sat E )
2 satffunlem2lem2.a 
 |-  A = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) )
3 satffunlem2lem2.b 
 |-  B = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) }
4 1 fveq1i 
 |-  ( S ` suc N ) = ( ( M Sat E ) ` suc N )
5 4 dmeqi 
 |-  dom ( S ` suc N ) = dom ( ( M Sat E ) ` suc N )
6 simprl 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> M e. V )
7 simprr 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> E e. W )
8 peano2 
 |-  ( N e. _om -> suc N e. _om )
9 8 adantr 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> suc N e. _om )
10 6 7 9 3jca 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> ( M e. V /\ E e. W /\ suc N e. _om ) )
11 satfdmfmla 
 |-  ( ( M e. V /\ E e. W /\ suc N e. _om ) -> dom ( ( M Sat E ) ` suc N ) = ( Fmla ` suc N ) )
12 10 11 syl 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> dom ( ( M Sat E ) ` suc N ) = ( Fmla ` suc N ) )
13 12 adantr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> dom ( ( M Sat E ) ` suc N ) = ( Fmla ` suc N ) )
14 5 13 eqtrid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> dom ( S ` suc N ) = ( Fmla ` suc N ) )
15 ovex 
 |-  ( M ^m _om ) e. _V
16 15 difexi 
 |-  ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) e. _V
17 2 16 eqeltri 
 |-  A e. _V
18 17 a1i 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` suc N ) ) /\ v e. ( S ` suc N ) ) -> A e. _V )
19 18 ralrimiva 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` suc N ) ) -> A. v e. ( S ` suc N ) A e. _V )
20 3 15 rabex2 
 |-  B e. _V
21 20 a1i 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` suc N ) ) /\ i e. _om ) -> B e. _V )
22 21 ralrimiva 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` suc N ) ) -> A. i e. _om B e. _V )
23 19 22 jca 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` suc N ) ) -> ( A. v e. ( S ` suc N ) A e. _V /\ A. i e. _om B e. _V ) )
24 23 ralrimiva 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> A. u e. ( S ` suc N ) ( A. v e. ( S ` suc N ) A e. _V /\ A. i e. _om B e. _V ) )
25 simplr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( M e. V /\ E e. W ) )
26 8 ancri 
 |-  ( N e. _om -> ( suc N e. _om /\ N e. _om ) )
27 26 ad2antrr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( suc N e. _om /\ N e. _om ) )
28 25 27 jca 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( M e. V /\ E e. W ) /\ ( suc N e. _om /\ N e. _om ) ) )
29 sssucid 
 |-  N C_ suc N
30 1 satfsschain 
 |-  ( ( ( M e. V /\ E e. W ) /\ ( suc N e. _om /\ N e. _om ) ) -> ( N C_ suc N -> ( S ` N ) C_ ( S ` suc N ) ) )
31 28 29 30 mpisyl 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( S ` N ) C_ ( S ` suc N ) )
32 dmopab3rexdif 
 |-  ( ( A. u e. ( S ` suc N ) ( A. v e. ( S ` suc N ) A e. _V /\ A. i e. _om B e. _V ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> dom { <. x , y >. | ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) ) } = { x | ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) } )
33 24 31 32 syl2anc 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> dom { <. x , y >. | ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) ) } = { x | ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) } )
34 simpr 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) )
35 fveqeq2 
 |-  ( w = u -> ( ( 1st ` w ) = ( 1st ` u ) <-> ( 1st ` u ) = ( 1st ` u ) ) )
36 35 adantl 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) /\ w = u ) -> ( ( 1st ` w ) = ( 1st ` u ) <-> ( 1st ` u ) = ( 1st ` u ) ) )
37 eqidd 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> ( 1st ` u ) = ( 1st ` u ) )
38 34 36 37 rspcedvd 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> E. w e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` w ) = ( 1st ` u ) )
39 4 funeqi 
 |-  ( Fun ( S ` suc N ) <-> Fun ( ( M Sat E ) ` suc N ) )
40 39 bilani 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> Fun ( ( M Sat E ) ` suc N ) )
41 1 fveq1i 
 |-  ( S ` N ) = ( ( M Sat E ) ` N )
42 31 41 4 3sstr3g 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( M Sat E ) ` N ) C_ ( ( M Sat E ) ` suc N ) )
43 40 42 jca 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( Fun ( ( M Sat E ) ` suc N ) /\ ( ( M Sat E ) ` N ) C_ ( ( M Sat E ) ` suc N ) ) )
44 43 adantr 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> ( Fun ( ( M Sat E ) ` suc N ) /\ ( ( M Sat E ) ` N ) C_ ( ( M Sat E ) ` suc N ) ) )
45 funeldmdif 
 |-  ( ( Fun ( ( M Sat E ) ` suc N ) /\ ( ( M Sat E ) ` N ) C_ ( ( M Sat E ) ` suc N ) ) -> ( ( 1st ` u ) e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) <-> E. w e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` w ) = ( 1st ` u ) ) )
46 44 45 syl 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> ( ( 1st ` u ) e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) <-> E. w e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` w ) = ( 1st ` u ) ) )
47 38 46 mpbird 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> ( 1st ` u ) e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) )
48 47 ex 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) -> ( 1st ` u ) e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) ) )
49 4 41 difeq12i 
 |-  ( ( S ` suc N ) \ ( S ` N ) ) = ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) )
50 49 eleq2i 
 |-  ( u e. ( ( S ` suc N ) \ ( S ` N ) ) <-> u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) )
51 50 a1i 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( u e. ( ( S ` suc N ) \ ( S ` N ) ) <-> u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) )
52 13 eqcomd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( Fmla ` suc N ) = dom ( ( M Sat E ) ` suc N ) )
53 simpl 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> N e. _om )
54 6 7 53 3jca 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> ( M e. V /\ E e. W /\ N e. _om ) )
55 satfdmfmla 
 |-  ( ( M e. V /\ E e. W /\ N e. _om ) -> dom ( ( M Sat E ) ` N ) = ( Fmla ` N ) )
56 54 55 syl 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> dom ( ( M Sat E ) ` N ) = ( Fmla ` N ) )
57 56 eqcomd 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> ( Fmla ` N ) = dom ( ( M Sat E ) ` N ) )
58 57 adantr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( Fmla ` N ) = dom ( ( M Sat E ) ` N ) )
59 52 58 difeq12d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) = ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) )
60 59 eleq2d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( 1st ` u ) e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) <-> ( 1st ` u ) e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) ) )
61 48 51 60 3imtr4d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( u e. ( ( S ` suc N ) \ ( S ` N ) ) -> ( 1st ` u ) e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ) )
62 61 imp 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( 1st ` u ) e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) )
63 62 adantr 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) -> ( 1st ` u ) e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) )
64 oveq1 
 |-  ( f = ( 1st ` u ) -> ( f |g g ) = ( ( 1st ` u ) |g g ) )
65 64 eqeq2d 
 |-  ( f = ( 1st ` u ) -> ( x = ( f |g g ) <-> x = ( ( 1st ` u ) |g g ) ) )
66 65 rexbidv 
 |-  ( f = ( 1st ` u ) -> ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) <-> E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) ) )
67 eqidd 
 |-  ( f = ( 1st ` u ) -> i = i )
68 id 
 |-  ( f = ( 1st ` u ) -> f = ( 1st ` u ) )
69 67 68 goaleq12d 
 |-  ( f = ( 1st ` u ) -> A.g i f = A.g i ( 1st ` u ) )
70 69 eqeq2d 
 |-  ( f = ( 1st ` u ) -> ( x = A.g i f <-> x = A.g i ( 1st ` u ) ) )
71 70 rexbidv 
 |-  ( f = ( 1st ` u ) -> ( E. i e. _om x = A.g i f <-> E. i e. _om x = A.g i ( 1st ` u ) ) )
72 66 71 orbi12d 
 |-  ( f = ( 1st ` u ) -> ( ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) <-> ( E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) )
73 72 adantl 
 |-  ( ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) /\ f = ( 1st ` u ) ) -> ( ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) <-> ( E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) )
74 6 adantr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> M e. V )
75 7 adantr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> E e. W )
76 8 ad2antrr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> suc N e. _om )
77 74 75 76 3jca 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( M e. V /\ E e. W /\ suc N e. _om ) )
78 satfrel 
 |-  ( ( M e. V /\ E e. W /\ suc N e. _om ) -> Rel ( ( M Sat E ) ` suc N ) )
79 77 78 syl 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> Rel ( ( M Sat E ) ` suc N ) )
80 4 releqi 
 |-  ( Rel ( S ` suc N ) <-> Rel ( ( M Sat E ) ` suc N ) )
81 79 80 sylibr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> Rel ( S ` suc N ) )
82 1stdm 
 |-  ( ( Rel ( S ` suc N ) /\ v e. ( S ` suc N ) ) -> ( 1st ` v ) e. dom ( S ` suc N ) )
83 81 82 sylan 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ v e. ( S ` suc N ) ) -> ( 1st ` v ) e. dom ( S ` suc N ) )
84 14 eqcomd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( Fmla ` suc N ) = dom ( S ` suc N ) )
85 84 adantr 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ v e. ( S ` suc N ) ) -> ( Fmla ` suc N ) = dom ( S ` suc N ) )
86 83 85 eleqtrrd 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ v e. ( S ` suc N ) ) -> ( 1st ` v ) e. ( Fmla ` suc N ) )
87 86 ad4ant13 
 |-  ( ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ v e. ( S ` suc N ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> ( 1st ` v ) e. ( Fmla ` suc N ) )
88 oveq2 
 |-  ( g = ( 1st ` v ) -> ( ( 1st ` u ) |g g ) = ( ( 1st ` u ) |g ( 1st ` v ) ) )
89 88 eqeq2d 
 |-  ( g = ( 1st ` v ) -> ( x = ( ( 1st ` u ) |g g ) <-> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
90 89 adantl 
 |-  ( ( ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ v e. ( S ` suc N ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) /\ g = ( 1st ` v ) ) -> ( x = ( ( 1st ` u ) |g g ) <-> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
91 simpr 
 |-  ( ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ v e. ( S ` suc N ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> x = ( ( 1st ` u ) |g ( 1st ` v ) ) )
92 87 90 91 rspcedvd 
 |-  ( ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ v e. ( S ` suc N ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) )
93 92 rexlimdva2 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) -> E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) ) )
94 93 orim1d 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) -> ( E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) )
95 94 imp 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) -> ( E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) )
96 63 73 95 rspcedvd 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) -> E. f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) )
97 96 rexlimdva2 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) -> E. f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) )
98 54 adantr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( M e. V /\ E e. W /\ N e. _om ) )
99 satfrel 
 |-  ( ( M e. V /\ E e. W /\ N e. _om ) -> Rel ( ( M Sat E ) ` N ) )
100 98 99 syl 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> Rel ( ( M Sat E ) ` N ) )
101 41 releqi 
 |-  ( Rel ( S ` N ) <-> Rel ( ( M Sat E ) ` N ) )
102 100 101 sylibr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> Rel ( S ` N ) )
103 1stdm 
 |-  ( ( Rel ( S ` N ) /\ u e. ( S ` N ) ) -> ( 1st ` u ) e. dom ( S ` N ) )
104 102 103 sylan 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) -> ( 1st ` u ) e. dom ( S ` N ) )
105 41 dmeqi 
 |-  dom ( S ` N ) = dom ( ( M Sat E ) ` N )
106 98 55 syl 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> dom ( ( M Sat E ) ` N ) = ( Fmla ` N ) )
107 105 106 eqtrid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> dom ( S ` N ) = ( Fmla ` N ) )
108 107 eqcomd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( Fmla ` N ) = dom ( S ` N ) )
109 108 adantr 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) -> ( Fmla ` N ) = dom ( S ` N ) )
110 104 109 eleqtrrd 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) -> ( 1st ` u ) e. ( Fmla ` N ) )
111 110 adantr 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) /\ E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> ( 1st ` u ) e. ( Fmla ` N ) )
112 65 rexbidv 
 |-  ( f = ( 1st ` u ) -> ( E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) <-> E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( ( 1st ` u ) |g g ) ) )
113 112 adantl 
 |-  ( ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) /\ E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) /\ f = ( 1st ` u ) ) -> ( E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) <-> E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( ( 1st ` u ) |g g ) ) )
114 simpr 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) )
115 fveqeq2 
 |-  ( t = v -> ( ( 1st ` t ) = ( 1st ` v ) <-> ( 1st ` v ) = ( 1st ` v ) ) )
116 115 adantl 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) /\ t = v ) -> ( ( 1st ` t ) = ( 1st ` v ) <-> ( 1st ` v ) = ( 1st ` v ) ) )
117 eqidd 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> ( 1st ` v ) = ( 1st ` v ) )
118 114 116 117 rspcedvd 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> E. t e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` t ) = ( 1st ` v ) )
119 43 adantr 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> ( Fun ( ( M Sat E ) ` suc N ) /\ ( ( M Sat E ) ` N ) C_ ( ( M Sat E ) ` suc N ) ) )
120 funeldmdif 
 |-  ( ( Fun ( ( M Sat E ) ` suc N ) /\ ( ( M Sat E ) ` N ) C_ ( ( M Sat E ) ` suc N ) ) -> ( ( 1st ` v ) e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) <-> E. t e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` t ) = ( 1st ` v ) ) )
121 119 120 syl 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> ( ( 1st ` v ) e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) <-> E. t e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` t ) = ( 1st ` v ) ) )
122 118 121 mpbird 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) -> ( 1st ` v ) e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) )
123 122 ex 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) -> ( 1st ` v ) e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) ) )
124 49 eleq2i 
 |-  ( v e. ( ( S ` suc N ) \ ( S ` N ) ) <-> v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) )
125 124 a1i 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( v e. ( ( S ` suc N ) \ ( S ` N ) ) <-> v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ) )
126 12 eqcomd 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> ( Fmla ` suc N ) = dom ( ( M Sat E ) ` suc N ) )
127 126 57 difeq12d 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) = ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) )
128 127 eleq2d 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> ( ( 1st ` v ) e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) <-> ( 1st ` v ) e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) ) )
129 128 adantr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( 1st ` v ) e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) <-> ( 1st ` v ) e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) ) )
130 123 125 129 3imtr4d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( v e. ( ( S ` suc N ) \ ( S ` N ) ) -> ( 1st ` v ) e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ) )
131 130 adantr 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) -> ( v e. ( ( S ` suc N ) \ ( S ` N ) ) -> ( 1st ` v ) e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ) )
132 131 imp 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( 1st ` v ) e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) )
133 132 adantr 
 |-  ( ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> ( 1st ` v ) e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) )
134 89 adantl 
 |-  ( ( ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) /\ g = ( 1st ` v ) ) -> ( x = ( ( 1st ` u ) |g g ) <-> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
135 simpr 
 |-  ( ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> x = ( ( 1st ` u ) |g ( 1st ` v ) ) )
136 133 134 135 rspcedvd 
 |-  ( ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( ( 1st ` u ) |g g ) )
137 136 r19.29an 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) /\ E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( ( 1st ` u ) |g g ) )
138 111 113 137 rspcedvd 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( S ` N ) ) /\ E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> E. f e. ( Fmla ` N ) E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) )
139 138 rexlimdva2 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) -> E. f e. ( Fmla ` N ) E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) )
140 97 139 orim12d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> ( E. f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) \/ E. f e. ( Fmla ` N ) E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) ) )
141 10 adantr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( M e. V /\ E e. W /\ suc N e. _om ) )
142 11 eqcomd 
 |-  ( ( M e. V /\ E e. W /\ suc N e. _om ) -> ( Fmla ` suc N ) = dom ( ( M Sat E ) ` suc N ) )
143 141 142 syl 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( Fmla ` suc N ) = dom ( ( M Sat E ) ` suc N ) )
144 106 eqcomd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( Fmla ` N ) = dom ( ( M Sat E ) ` N ) )
145 143 144 difeq12d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) = ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) )
146 145 eleq2d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) <-> f e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) ) )
147 eqid 
 |-  ( M Sat E ) = ( M Sat E )
148 147 satfsschain 
 |-  ( ( ( M e. V /\ E e. W ) /\ ( suc N e. _om /\ N e. _om ) ) -> ( N C_ suc N -> ( ( M Sat E ) ` N ) C_ ( ( M Sat E ) ` suc N ) ) )
149 28 29 148 mpisyl 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( M Sat E ) ` N ) C_ ( ( M Sat E ) ` suc N ) )
150 releldmdifi 
 |-  ( ( Rel ( ( M Sat E ) ` suc N ) /\ ( ( M Sat E ) ` N ) C_ ( ( M Sat E ) ` suc N ) ) -> ( f e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) -> E. u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` u ) = f ) )
151 79 149 150 syl2anc 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( f e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) -> E. u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` u ) = f ) )
152 146 151 sylbid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) -> E. u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` u ) = f ) )
153 49 eqcomi 
 |-  ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) = ( ( S ` suc N ) \ ( S ` N ) )
154 153 rexeqi 
 |-  ( E. u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` u ) = f <-> E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( 1st ` u ) = f )
155 r19.41v 
 |-  ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( ( 1st ` u ) = f /\ ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) <-> ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( 1st ` u ) = f /\ ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) )
156 oveq1 
 |-  ( ( 1st ` u ) = f -> ( ( 1st ` u ) |g g ) = ( f |g g ) )
157 156 eqeq2d 
 |-  ( ( 1st ` u ) = f -> ( x = ( ( 1st ` u ) |g g ) <-> x = ( f |g g ) ) )
158 157 rexbidv 
 |-  ( ( 1st ` u ) = f -> ( E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) <-> E. g e. ( Fmla ` suc N ) x = ( f |g g ) ) )
159 eqidd 
 |-  ( ( 1st ` u ) = f -> i = i )
160 id 
 |-  ( ( 1st ` u ) = f -> ( 1st ` u ) = f )
161 159 160 goaleq12d 
 |-  ( ( 1st ` u ) = f -> A.g i ( 1st ` u ) = A.g i f )
162 161 eqeq2d 
 |-  ( ( 1st ` u ) = f -> ( x = A.g i ( 1st ` u ) <-> x = A.g i f ) )
163 162 rexbidv 
 |-  ( ( 1st ` u ) = f -> ( E. i e. _om x = A.g i ( 1st ` u ) <-> E. i e. _om x = A.g i f ) )
164 158 163 orbi12d 
 |-  ( ( 1st ` u ) = f -> ( ( E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) <-> ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) )
165 164 adantl 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ ( 1st ` u ) = f ) -> ( ( E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) <-> ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) )
166 141 11 syl 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> dom ( ( M Sat E ) ` suc N ) = ( Fmla ` suc N ) )
167 166 eqcomd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( Fmla ` suc N ) = dom ( ( M Sat E ) ` suc N ) )
168 167 eleq2d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( g e. ( Fmla ` suc N ) <-> g e. dom ( ( M Sat E ) ` suc N ) ) )
169 releldm2 
 |-  ( Rel ( ( M Sat E ) ` suc N ) -> ( g e. dom ( ( M Sat E ) ` suc N ) <-> E. v e. ( ( M Sat E ) ` suc N ) ( 1st ` v ) = g ) )
170 79 169 syl 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( g e. dom ( ( M Sat E ) ` suc N ) <-> E. v e. ( ( M Sat E ) ` suc N ) ( 1st ` v ) = g ) )
171 168 170 bitrd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( g e. ( Fmla ` suc N ) <-> E. v e. ( ( M Sat E ) ` suc N ) ( 1st ` v ) = g ) )
172 r19.41v 
 |-  ( E. v e. ( ( M Sat E ) ` suc N ) ( ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) <-> ( E. v e. ( ( M Sat E ) ` suc N ) ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) )
173 1 eqcomi 
 |-  ( M Sat E ) = S
174 173 fveq1i 
 |-  ( ( M Sat E ) ` suc N ) = ( S ` suc N )
175 174 rexeqi 
 |-  ( E. v e. ( ( M Sat E ) ` suc N ) ( ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) <-> E. v e. ( S ` suc N ) ( ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) )
176 88 eqcoms 
 |-  ( ( 1st ` v ) = g -> ( ( 1st ` u ) |g g ) = ( ( 1st ` u ) |g ( 1st ` v ) ) )
177 176 eqeq2d 
 |-  ( ( 1st ` v ) = g -> ( x = ( ( 1st ` u ) |g g ) <-> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
178 177 biimpa 
 |-  ( ( ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) -> x = ( ( 1st ` u ) |g ( 1st ` v ) ) )
179 178 a1i 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) -> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
180 179 reximdv 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. v e. ( S ` suc N ) ( ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) -> E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
181 175 180 biimtrid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. v e. ( ( M Sat E ) ` suc N ) ( ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) -> E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
182 172 181 biimtrrid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( E. v e. ( ( M Sat E ) ` suc N ) ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) -> E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
183 182 expd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. v e. ( ( M Sat E ) ` suc N ) ( 1st ` v ) = g -> ( x = ( ( 1st ` u ) |g g ) -> E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) )
184 171 183 sylbid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( g e. ( Fmla ` suc N ) -> ( x = ( ( 1st ` u ) |g g ) -> E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) )
185 184 rexlimdv 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) -> E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
186 185 ad2antrr 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ ( 1st ` u ) = f ) -> ( E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) -> E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
187 186 orim1d 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ ( 1st ` u ) = f ) -> ( ( E. g e. ( Fmla ` suc N ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) -> ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) )
188 165 187 sylbird 
 |-  ( ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ ( 1st ` u ) = f ) -> ( ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) -> ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) )
189 188 expimpd 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( ( 1st ` u ) = f /\ ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) -> ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) )
190 189 reximdva 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( ( 1st ` u ) = f /\ ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) -> E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) )
191 155 190 biimtrrid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( 1st ` u ) = f /\ ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) -> E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) )
192 191 expd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( 1st ` u ) = f -> ( ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) -> E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) )
193 154 192 biimtrid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. u e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` u ) = f -> ( ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) -> E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) )
194 152 193 syld 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) -> ( ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) -> E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) )
195 194 rexlimdv 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) -> E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) )
196 144 eleq2d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( f e. ( Fmla ` N ) <-> f e. dom ( ( M Sat E ) ` N ) ) )
197 54 99 syl 
 |-  ( ( N e. _om /\ ( M e. V /\ E e. W ) ) -> Rel ( ( M Sat E ) ` N ) )
198 197 adantr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> Rel ( ( M Sat E ) ` N ) )
199 releldm2 
 |-  ( Rel ( ( M Sat E ) ` N ) -> ( f e. dom ( ( M Sat E ) ` N ) <-> E. u e. ( ( M Sat E ) ` N ) ( 1st ` u ) = f ) )
200 198 199 syl 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( f e. dom ( ( M Sat E ) ` N ) <-> E. u e. ( ( M Sat E ) ` N ) ( 1st ` u ) = f ) )
201 196 200 bitrd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( f e. ( Fmla ` N ) <-> E. u e. ( ( M Sat E ) ` N ) ( 1st ` u ) = f ) )
202 r19.41v 
 |-  ( E. u e. ( ( M Sat E ) ` N ) ( ( 1st ` u ) = f /\ E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) <-> ( E. u e. ( ( M Sat E ) ` N ) ( 1st ` u ) = f /\ E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) )
203 41 eqcomi 
 |-  ( ( M Sat E ) ` N ) = ( S ` N )
204 203 rexeqi 
 |-  ( E. u e. ( ( M Sat E ) ` N ) ( ( 1st ` u ) = f /\ E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) <-> E. u e. ( S ` N ) ( ( 1st ` u ) = f /\ E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) )
205 157 rexbidv 
 |-  ( ( 1st ` u ) = f -> ( E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( ( 1st ` u ) |g g ) <-> E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) )
206 205 adantl 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ ( 1st ` u ) = f ) -> ( E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( ( 1st ` u ) |g g ) <-> E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) )
207 145 eleq2d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) <-> g e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) ) )
208 releldmdifi 
 |-  ( ( Rel ( ( M Sat E ) ` suc N ) /\ ( ( M Sat E ) ` N ) C_ ( ( M Sat E ) ` suc N ) ) -> ( g e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) -> E. v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` v ) = g ) )
209 79 149 208 syl2anc 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( g e. ( dom ( ( M Sat E ) ` suc N ) \ dom ( ( M Sat E ) ` N ) ) -> E. v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` v ) = g ) )
210 207 209 sylbid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) -> E. v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` v ) = g ) )
211 153 rexeqi 
 |-  ( E. v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` v ) = g <-> E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( 1st ` v ) = g )
212 177 biimpcd 
 |-  ( x = ( ( 1st ` u ) |g g ) -> ( ( 1st ` v ) = g -> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
213 212 adantl 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ x = ( ( 1st ` u ) |g g ) ) -> ( ( 1st ` v ) = g -> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
214 213 reximdv 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ x = ( ( 1st ` u ) |g g ) ) -> ( E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( 1st ` v ) = g -> E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
215 214 ex 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( x = ( ( 1st ` u ) |g g ) -> ( E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( 1st ` v ) = g -> E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) )
216 215 com23 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( 1st ` v ) = g -> ( x = ( ( 1st ` u ) |g g ) -> E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) )
217 211 216 biimtrid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. v e. ( ( ( M Sat E ) ` suc N ) \ ( ( M Sat E ) ` N ) ) ( 1st ` v ) = g -> ( x = ( ( 1st ` u ) |g g ) -> E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) )
218 210 217 syld 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) -> ( x = ( ( 1st ` u ) |g g ) -> E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) )
219 218 rexlimdv 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( ( 1st ` u ) |g g ) -> E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
220 219 adantr 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ ( 1st ` u ) = f ) -> ( E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( ( 1st ` u ) |g g ) -> E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
221 206 220 sylbird 
 |-  ( ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) /\ ( 1st ` u ) = f ) -> ( E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) -> E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
222 221 expimpd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( ( 1st ` u ) = f /\ E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) -> E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
223 222 reximdv 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. u e. ( S ` N ) ( ( 1st ` u ) = f /\ E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) -> E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
224 204 223 biimtrid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. u e. ( ( M Sat E ) ` N ) ( ( 1st ` u ) = f /\ E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) -> E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
225 202 224 biimtrrid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( E. u e. ( ( M Sat E ) ` N ) ( 1st ` u ) = f /\ E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) -> E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
226 225 expd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. u e. ( ( M Sat E ) ` N ) ( 1st ` u ) = f -> ( E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) -> E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) )
227 201 226 sylbid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( f e. ( Fmla ` N ) -> ( E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) -> E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) )
228 227 rexlimdv 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( E. f e. ( Fmla ` N ) E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) -> E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) )
229 195 228 orim12d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( E. f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) \/ E. f e. ( Fmla ` N ) E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) -> ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) )
230 140 229 impbid 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) <-> ( E. f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) \/ E. f e. ( Fmla ` N ) E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) ) )
231 230 abbidv 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> { x | ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) } = { x | ( E. f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) \/ E. f e. ( Fmla ` N ) E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) } )
232 33 231 eqtrd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> dom { <. x , y >. | ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) ) } = { x | ( E. f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) \/ E. f e. ( Fmla ` N ) E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) } )
233 14 232 ineq12d 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( dom ( S ` suc N ) i^i dom { <. x , y >. | ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) ) } ) = ( ( Fmla ` suc N ) i^i { x | ( E. f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) \/ E. f e. ( Fmla ` N ) E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) } ) )
234 fmlasucdisj 
 |-  ( N e. _om -> ( ( Fmla ` suc N ) i^i { x | ( E. f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) \/ E. f e. ( Fmla ` N ) E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) } ) = (/) )
235 234 ad2antrr 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( ( Fmla ` suc N ) i^i { x | ( E. f e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) ( E. g e. ( Fmla ` suc N ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) \/ E. f e. ( Fmla ` N ) E. g e. ( ( Fmla ` suc N ) \ ( Fmla ` N ) ) x = ( f |g g ) ) } ) = (/) )
236 233 235 eqtrd 
 |-  ( ( ( N e. _om /\ ( M e. V /\ E e. W ) ) /\ Fun ( S ` suc N ) ) -> ( dom ( S ` suc N ) i^i dom { <. x , y >. | ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) ) } ) = (/) )