Step |
Hyp |
Ref |
Expression |
1 |
|
peano1 |
|- (/) e. _om |
2 |
|
satfdmfmla |
|- ( ( M e. V /\ E e. W /\ (/) e. _om ) -> dom ( ( M Sat E ) ` (/) ) = ( Fmla ` (/) ) ) |
3 |
1 2
|
mp3an3 |
|- ( ( M e. V /\ E e. W ) -> dom ( ( M Sat E ) ` (/) ) = ( Fmla ` (/) ) ) |
4 |
|
ovex |
|- ( M ^m _om ) e. _V |
5 |
4
|
difexi |
|- ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) e. _V |
6 |
5
|
a1i |
|- ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ v e. ( ( M Sat E ) ` (/) ) ) -> ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) e. _V ) |
7 |
6
|
ralrimiva |
|- ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> A. v e. ( ( M Sat E ) ` (/) ) ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) e. _V ) |
8 |
4
|
rabex |
|- { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } e. _V |
9 |
8
|
a1i |
|- ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ i e. _om ) -> { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } e. _V ) |
10 |
9
|
ralrimiva |
|- ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> A. i e. _om { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } e. _V ) |
11 |
7 10
|
jca |
|- ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> ( A. v e. ( ( M Sat E ) ` (/) ) ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) e. _V /\ A. i e. _om { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } e. _V ) ) |
12 |
11
|
ralrimiva |
|- ( ( M e. V /\ E e. W ) -> A. u e. ( ( M Sat E ) ` (/) ) ( A. v e. ( ( M Sat E ) ` (/) ) ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) e. _V /\ A. i e. _om { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } e. _V ) ) |
13 |
|
dmopab2rex |
|- ( A. u e. ( ( M Sat E ) ` (/) ) ( A. v e. ( ( M Sat E ) ` (/) ) ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) e. _V /\ A. i e. _om { f e. ( M ^m _om ) | A. j e. M ( { <. i , j >. } u. ( f |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } e. _V ) -> 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 ) } ) ) } = { x | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) } ) |
14 |
12 13
|
syl |
|- ( ( M e. V /\ E e. W ) -> 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 ) } ) ) } = { x | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) } ) |
15 |
|
satfrel |
|- ( ( M e. V /\ E e. W /\ (/) e. _om ) -> Rel ( ( M Sat E ) ` (/) ) ) |
16 |
1 15
|
mp3an3 |
|- ( ( M e. V /\ E e. W ) -> Rel ( ( M Sat E ) ` (/) ) ) |
17 |
|
1stdm |
|- ( ( Rel ( ( M Sat E ) ` (/) ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> ( 1st ` u ) e. dom ( ( M Sat E ) ` (/) ) ) |
18 |
16 17
|
sylan |
|- ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> ( 1st ` u ) e. dom ( ( M Sat E ) ` (/) ) ) |
19 |
2
|
eqcomd |
|- ( ( M e. V /\ E e. W /\ (/) e. _om ) -> ( Fmla ` (/) ) = dom ( ( M Sat E ) ` (/) ) ) |
20 |
1 19
|
mp3an3 |
|- ( ( M e. V /\ E e. W ) -> ( Fmla ` (/) ) = dom ( ( M Sat E ) ` (/) ) ) |
21 |
20
|
adantr |
|- ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> ( Fmla ` (/) ) = dom ( ( M Sat E ) ` (/) ) ) |
22 |
18 21
|
eleqtrrd |
|- ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> ( 1st ` u ) e. ( Fmla ` (/) ) ) |
23 |
22
|
adantr |
|- ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) -> ( 1st ` u ) e. ( Fmla ` (/) ) ) |
24 |
|
oveq1 |
|- ( f = ( 1st ` u ) -> ( f |g g ) = ( ( 1st ` u ) |g g ) ) |
25 |
24
|
eqeq2d |
|- ( f = ( 1st ` u ) -> ( x = ( f |g g ) <-> x = ( ( 1st ` u ) |g g ) ) ) |
26 |
25
|
rexbidv |
|- ( f = ( 1st ` u ) -> ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) <-> E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) ) ) |
27 |
|
eqidd |
|- ( f = ( 1st ` u ) -> i = i ) |
28 |
|
id |
|- ( f = ( 1st ` u ) -> f = ( 1st ` u ) ) |
29 |
27 28
|
goaleq12d |
|- ( f = ( 1st ` u ) -> A.g i f = A.g i ( 1st ` u ) ) |
30 |
29
|
eqeq2d |
|- ( f = ( 1st ` u ) -> ( x = A.g i f <-> x = A.g i ( 1st ` u ) ) ) |
31 |
30
|
rexbidv |
|- ( f = ( 1st ` u ) -> ( E. i e. _om x = A.g i f <-> E. i e. _om x = A.g i ( 1st ` u ) ) ) |
32 |
26 31
|
orbi12d |
|- ( f = ( 1st ` u ) -> ( ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) <-> ( E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
33 |
32
|
adantl |
|- ( ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) /\ f = ( 1st ` u ) ) -> ( ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) <-> ( E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
34 |
|
1stdm |
|- ( ( Rel ( ( M Sat E ) ` (/) ) /\ v e. ( ( M Sat E ) ` (/) ) ) -> ( 1st ` v ) e. dom ( ( M Sat E ) ` (/) ) ) |
35 |
16 34
|
sylan |
|- ( ( ( M e. V /\ E e. W ) /\ v e. ( ( M Sat E ) ` (/) ) ) -> ( 1st ` v ) e. dom ( ( M Sat E ) ` (/) ) ) |
36 |
20
|
adantr |
|- ( ( ( M e. V /\ E e. W ) /\ v e. ( ( M Sat E ) ` (/) ) ) -> ( Fmla ` (/) ) = dom ( ( M Sat E ) ` (/) ) ) |
37 |
35 36
|
eleqtrrd |
|- ( ( ( M e. V /\ E e. W ) /\ v e. ( ( M Sat E ) ` (/) ) ) -> ( 1st ` v ) e. ( Fmla ` (/) ) ) |
38 |
37
|
ad4ant13 |
|- ( ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ v e. ( ( M Sat E ) ` (/) ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> ( 1st ` v ) e. ( Fmla ` (/) ) ) |
39 |
|
oveq2 |
|- ( g = ( 1st ` v ) -> ( ( 1st ` u ) |g g ) = ( ( 1st ` u ) |g ( 1st ` v ) ) ) |
40 |
39
|
eqeq2d |
|- ( g = ( 1st ` v ) -> ( x = ( ( 1st ` u ) |g g ) <-> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
41 |
40
|
adantl |
|- ( ( ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ v e. ( ( M Sat E ) ` (/) ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) /\ g = ( 1st ` v ) ) -> ( x = ( ( 1st ` u ) |g g ) <-> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
42 |
|
simpr |
|- ( ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ v e. ( ( M Sat E ) ` (/) ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) |
43 |
38 41 42
|
rspcedvd |
|- ( ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ v e. ( ( M Sat E ) ` (/) ) ) /\ x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) -> E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) ) |
44 |
43
|
ex |
|- ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ v e. ( ( M Sat E ) ` (/) ) ) -> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) -> E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) ) ) |
45 |
44
|
rexlimdva |
|- ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) -> E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) ) ) |
46 |
45
|
orim1d |
|- ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> ( ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) -> ( E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
47 |
46
|
imp |
|- ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) -> ( E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) |
48 |
23 33 47
|
rspcedvd |
|- ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) -> E. f e. ( Fmla ` (/) ) ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) |
49 |
48
|
ex |
|- ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> ( ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) -> E. f e. ( Fmla ` (/) ) ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) ) |
50 |
49
|
rexlimdva |
|- ( ( M e. V /\ E e. W ) -> ( E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) -> E. f e. ( Fmla ` (/) ) ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) ) |
51 |
|
releldm2 |
|- ( Rel ( ( M Sat E ) ` (/) ) -> ( f e. dom ( ( M Sat E ) ` (/) ) <-> E. u e. ( ( M Sat E ) ` (/) ) ( 1st ` u ) = f ) ) |
52 |
16 51
|
syl |
|- ( ( M e. V /\ E e. W ) -> ( f e. dom ( ( M Sat E ) ` (/) ) <-> E. u e. ( ( M Sat E ) ` (/) ) ( 1st ` u ) = f ) ) |
53 |
3
|
eleq2d |
|- ( ( M e. V /\ E e. W ) -> ( f e. dom ( ( M Sat E ) ` (/) ) <-> f e. ( Fmla ` (/) ) ) ) |
54 |
52 53
|
bitr3d |
|- ( ( M e. V /\ E e. W ) -> ( E. u e. ( ( M Sat E ) ` (/) ) ( 1st ` u ) = f <-> f e. ( Fmla ` (/) ) ) ) |
55 |
|
r19.41v |
|- ( E. u e. ( ( M Sat E ) ` (/) ) ( ( 1st ` u ) = f /\ ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) <-> ( E. u e. ( ( M Sat E ) ` (/) ) ( 1st ` u ) = f /\ ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) ) |
56 |
|
oveq1 |
|- ( ( 1st ` u ) = f -> ( ( 1st ` u ) |g g ) = ( f |g g ) ) |
57 |
56
|
eqeq2d |
|- ( ( 1st ` u ) = f -> ( x = ( ( 1st ` u ) |g g ) <-> x = ( f |g g ) ) ) |
58 |
57
|
rexbidv |
|- ( ( 1st ` u ) = f -> ( E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) <-> E. g e. ( Fmla ` (/) ) x = ( f |g g ) ) ) |
59 |
|
eqidd |
|- ( ( 1st ` u ) = f -> i = i ) |
60 |
|
id |
|- ( ( 1st ` u ) = f -> ( 1st ` u ) = f ) |
61 |
59 60
|
goaleq12d |
|- ( ( 1st ` u ) = f -> A.g i ( 1st ` u ) = A.g i f ) |
62 |
61
|
eqeq2d |
|- ( ( 1st ` u ) = f -> ( x = A.g i ( 1st ` u ) <-> x = A.g i f ) ) |
63 |
62
|
rexbidv |
|- ( ( 1st ` u ) = f -> ( E. i e. _om x = A.g i ( 1st ` u ) <-> E. i e. _om x = A.g i f ) ) |
64 |
58 63
|
orbi12d |
|- ( ( 1st ` u ) = f -> ( ( E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) <-> ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) ) |
65 |
64
|
adantl |
|- ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ ( 1st ` u ) = f ) -> ( ( E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) <-> ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) ) |
66 |
3
|
eqcomd |
|- ( ( M e. V /\ E e. W ) -> ( Fmla ` (/) ) = dom ( ( M Sat E ) ` (/) ) ) |
67 |
66
|
eleq2d |
|- ( ( M e. V /\ E e. W ) -> ( g e. ( Fmla ` (/) ) <-> g e. dom ( ( M Sat E ) ` (/) ) ) ) |
68 |
|
releldm2 |
|- ( Rel ( ( M Sat E ) ` (/) ) -> ( g e. dom ( ( M Sat E ) ` (/) ) <-> E. v e. ( ( M Sat E ) ` (/) ) ( 1st ` v ) = g ) ) |
69 |
16 68
|
syl |
|- ( ( M e. V /\ E e. W ) -> ( g e. dom ( ( M Sat E ) ` (/) ) <-> E. v e. ( ( M Sat E ) ` (/) ) ( 1st ` v ) = g ) ) |
70 |
67 69
|
bitrd |
|- ( ( M e. V /\ E e. W ) -> ( g e. ( Fmla ` (/) ) <-> E. v e. ( ( M Sat E ) ` (/) ) ( 1st ` v ) = g ) ) |
71 |
|
r19.41v |
|- ( E. v e. ( ( M Sat E ) ` (/) ) ( ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) <-> ( E. v e. ( ( M Sat E ) ` (/) ) ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) ) |
72 |
39
|
eqcoms |
|- ( ( 1st ` v ) = g -> ( ( 1st ` u ) |g g ) = ( ( 1st ` u ) |g ( 1st ` v ) ) ) |
73 |
72
|
eqeq2d |
|- ( ( 1st ` v ) = g -> ( x = ( ( 1st ` u ) |g g ) <-> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
74 |
73
|
biimpa |
|- ( ( ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) -> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) |
75 |
74
|
a1i |
|- ( ( M e. V /\ E e. W ) -> ( ( ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) -> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
76 |
75
|
reximdv |
|- ( ( M e. V /\ E e. W ) -> ( E. v e. ( ( M Sat E ) ` (/) ) ( ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) -> E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
77 |
71 76
|
syl5bir |
|- ( ( M e. V /\ E e. W ) -> ( ( E. v e. ( ( M Sat E ) ` (/) ) ( 1st ` v ) = g /\ x = ( ( 1st ` u ) |g g ) ) -> E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
78 |
77
|
expd |
|- ( ( M e. V /\ E e. W ) -> ( E. v e. ( ( M Sat E ) ` (/) ) ( 1st ` v ) = g -> ( x = ( ( 1st ` u ) |g g ) -> E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) ) |
79 |
70 78
|
sylbid |
|- ( ( M e. V /\ E e. W ) -> ( g e. ( Fmla ` (/) ) -> ( x = ( ( 1st ` u ) |g g ) -> E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) ) |
80 |
79
|
rexlimdv |
|- ( ( M e. V /\ E e. W ) -> ( E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) -> E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
81 |
80
|
adantr |
|- ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> ( E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) -> E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
82 |
81
|
adantr |
|- ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ ( 1st ` u ) = f ) -> ( E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) -> E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
83 |
82
|
orim1d |
|- ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ ( 1st ` u ) = f ) -> ( ( E. g e. ( Fmla ` (/) ) x = ( ( 1st ` u ) |g g ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) -> ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
84 |
65 83
|
sylbird |
|- ( ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) /\ ( 1st ` u ) = f ) -> ( ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) -> ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
85 |
84
|
expimpd |
|- ( ( ( M e. V /\ E e. W ) /\ u e. ( ( M Sat E ) ` (/) ) ) -> ( ( ( 1st ` u ) = f /\ ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) -> ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
86 |
85
|
reximdva |
|- ( ( M e. V /\ E e. W ) -> ( E. u e. ( ( M Sat E ) ` (/) ) ( ( 1st ` u ) = f /\ ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) -> E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
87 |
55 86
|
syl5bir |
|- ( ( M e. V /\ E e. W ) -> ( ( E. u e. ( ( M Sat E ) ` (/) ) ( 1st ` u ) = f /\ ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) -> E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
88 |
87
|
expd |
|- ( ( M e. V /\ E e. W ) -> ( E. u e. ( ( M Sat E ) ` (/) ) ( 1st ` u ) = f -> ( ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) -> E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) |
89 |
54 88
|
sylbird |
|- ( ( M e. V /\ E e. W ) -> ( f e. ( Fmla ` (/) ) -> ( ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) -> E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) |
90 |
89
|
rexlimdv |
|- ( ( M e. V /\ E e. W ) -> ( E. f e. ( Fmla ` (/) ) ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) -> E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
91 |
50 90
|
impbid |
|- ( ( M e. V /\ E e. W ) -> ( E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) <-> E. f e. ( Fmla ` (/) ) ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) ) ) |
92 |
91
|
abbidv |
|- ( ( M e. V /\ E e. W ) -> { x | E. u e. ( ( M Sat E ) ` (/) ) ( E. v e. ( ( M Sat E ) ` (/) ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) } = { x | E. f e. ( Fmla ` (/) ) ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) } ) |
93 |
14 92
|
eqtrd |
|- ( ( M e. V /\ E e. W ) -> 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 ) } ) ) } = { x | E. f e. ( Fmla ` (/) ) ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) } ) |
94 |
3 93
|
ineq12d |
|- ( ( 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 ) } ) ) } ) = ( ( Fmla ` (/) ) i^i { x | E. f e. ( Fmla ` (/) ) ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) } ) ) |
95 |
|
fmla0disjsuc |
|- ( ( Fmla ` (/) ) i^i { x | E. f e. ( Fmla ` (/) ) ( E. g e. ( Fmla ` (/) ) x = ( f |g g ) \/ E. i e. _om x = A.g i f ) } ) = (/) |
96 |
94 95
|
eqtrdi |
|- ( ( 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 ) } ) ) } ) = (/) ) |