Step |
Hyp |
Ref |
Expression |
1 |
|
satffunlem2lem1.s |
|- S = ( M Sat E ) |
2 |
|
satffunlem2lem1.a |
|- A = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) |
3 |
|
satffunlem2lem1.b |
|- B = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } |
4 |
|
simpl |
|- ( ( u = s /\ v = r ) -> u = s ) |
5 |
4
|
fveq2d |
|- ( ( u = s /\ v = r ) -> ( 1st ` u ) = ( 1st ` s ) ) |
6 |
|
simpr |
|- ( ( u = s /\ v = r ) -> v = r ) |
7 |
6
|
fveq2d |
|- ( ( u = s /\ v = r ) -> ( 1st ` v ) = ( 1st ` r ) ) |
8 |
5 7
|
oveq12d |
|- ( ( u = s /\ v = r ) -> ( ( 1st ` u ) |g ( 1st ` v ) ) = ( ( 1st ` s ) |g ( 1st ` r ) ) ) |
9 |
8
|
eqeq2d |
|- ( ( u = s /\ v = r ) -> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) <-> x = ( ( 1st ` s ) |g ( 1st ` r ) ) ) ) |
10 |
4
|
fveq2d |
|- ( ( u = s /\ v = r ) -> ( 2nd ` u ) = ( 2nd ` s ) ) |
11 |
6
|
fveq2d |
|- ( ( u = s /\ v = r ) -> ( 2nd ` v ) = ( 2nd ` r ) ) |
12 |
10 11
|
ineq12d |
|- ( ( u = s /\ v = r ) -> ( ( 2nd ` u ) i^i ( 2nd ` v ) ) = ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) |
13 |
12
|
difeq2d |
|- ( ( u = s /\ v = r ) -> ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) |
14 |
2 13
|
syl5eq |
|- ( ( u = s /\ v = r ) -> A = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) |
15 |
14
|
eqeq2d |
|- ( ( u = s /\ v = r ) -> ( y = A <-> y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) |
16 |
9 15
|
anbi12d |
|- ( ( u = s /\ v = r ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) <-> ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) ) |
17 |
16
|
cbvrexdva |
|- ( u = s -> ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) <-> E. r e. ( S ` suc N ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) ) |
18 |
|
simpr |
|- ( ( u = s /\ i = j ) -> i = j ) |
19 |
|
fveq2 |
|- ( u = s -> ( 1st ` u ) = ( 1st ` s ) ) |
20 |
19
|
adantr |
|- ( ( u = s /\ i = j ) -> ( 1st ` u ) = ( 1st ` s ) ) |
21 |
18 20
|
goaleq12d |
|- ( ( u = s /\ i = j ) -> A.g i ( 1st ` u ) = A.g j ( 1st ` s ) ) |
22 |
21
|
eqeq2d |
|- ( ( u = s /\ i = j ) -> ( x = A.g i ( 1st ` u ) <-> x = A.g j ( 1st ` s ) ) ) |
23 |
3
|
eqeq2i |
|- ( y = B <-> y = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) |
24 |
|
opeq1 |
|- ( i = j -> <. i , z >. = <. j , z >. ) |
25 |
24
|
sneqd |
|- ( i = j -> { <. i , z >. } = { <. j , z >. } ) |
26 |
|
sneq |
|- ( i = j -> { i } = { j } ) |
27 |
26
|
difeq2d |
|- ( i = j -> ( _om \ { i } ) = ( _om \ { j } ) ) |
28 |
27
|
reseq2d |
|- ( i = j -> ( a |` ( _om \ { i } ) ) = ( a |` ( _om \ { j } ) ) ) |
29 |
25 28
|
uneq12d |
|- ( i = j -> ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) = ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) ) |
30 |
29
|
adantl |
|- ( ( u = s /\ i = j ) -> ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) = ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) ) |
31 |
|
fveq2 |
|- ( u = s -> ( 2nd ` u ) = ( 2nd ` s ) ) |
32 |
31
|
adantr |
|- ( ( u = s /\ i = j ) -> ( 2nd ` u ) = ( 2nd ` s ) ) |
33 |
30 32
|
eleq12d |
|- ( ( u = s /\ i = j ) -> ( ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) <-> ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) ) ) |
34 |
33
|
ralbidv |
|- ( ( u = s /\ i = j ) -> ( A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) <-> A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) ) ) |
35 |
34
|
rabbidv |
|- ( ( u = s /\ i = j ) -> { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) |
36 |
35
|
eqeq2d |
|- ( ( u = s /\ i = j ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } <-> y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) ) |
37 |
23 36
|
syl5bb |
|- ( ( u = s /\ i = j ) -> ( y = B <-> y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) ) |
38 |
22 37
|
anbi12d |
|- ( ( u = s /\ i = j ) -> ( ( x = A.g i ( 1st ` u ) /\ y = B ) <-> ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) ) ) |
39 |
38
|
cbvrexdva |
|- ( u = s -> ( E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = B ) <-> E. j e. _om ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) ) ) |
40 |
17 39
|
orbi12d |
|- ( u = s -> ( ( 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. r e. ( S ` suc N ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) \/ E. j e. _om ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) ) ) ) |
41 |
40
|
cbvrexvw |
|- ( 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. s e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. r e. ( S ` suc N ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) \/ E. j e. _om ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) ) ) |
42 |
|
fveq2 |
|- ( v = r -> ( 1st ` v ) = ( 1st ` r ) ) |
43 |
19 42
|
oveqan12d |
|- ( ( u = s /\ v = r ) -> ( ( 1st ` u ) |g ( 1st ` v ) ) = ( ( 1st ` s ) |g ( 1st ` r ) ) ) |
44 |
43
|
eqeq2d |
|- ( ( u = s /\ v = r ) -> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) <-> x = ( ( 1st ` s ) |g ( 1st ` r ) ) ) ) |
45 |
2
|
eqeq2i |
|- ( y = A <-> y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) |
46 |
|
fveq2 |
|- ( v = r -> ( 2nd ` v ) = ( 2nd ` r ) ) |
47 |
31 46
|
ineqan12d |
|- ( ( u = s /\ v = r ) -> ( ( 2nd ` u ) i^i ( 2nd ` v ) ) = ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) |
48 |
47
|
difeq2d |
|- ( ( u = s /\ v = r ) -> ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) |
49 |
48
|
eqeq2d |
|- ( ( u = s /\ v = r ) -> ( y = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) <-> y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) |
50 |
45 49
|
syl5bb |
|- ( ( u = s /\ v = r ) -> ( y = A <-> y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) |
51 |
44 50
|
anbi12d |
|- ( ( u = s /\ v = r ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) <-> ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) ) |
52 |
51
|
cbvrexdva |
|- ( u = s -> ( E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) <-> E. r e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) ) |
53 |
52
|
cbvrexvw |
|- ( E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) <-> E. s e. ( S ` N ) E. r e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) |
54 |
41 53
|
orbi12i |
|- ( ( 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 ) ) <-> ( E. s e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. r e. ( S ` suc N ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) \/ E. j e. _om ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) ) \/ E. s e. ( S ` N ) E. r e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) ) |
55 |
|
simp-5l |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ v e. ( S ` suc N ) ) -> Fun ( S ` suc N ) ) |
56 |
|
eldifi |
|- ( s e. ( ( S ` suc N ) \ ( S ` N ) ) -> s e. ( S ` suc N ) ) |
57 |
56
|
adantl |
|- ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> s e. ( S ` suc N ) ) |
58 |
57
|
anim1i |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) -> ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) ) |
59 |
58
|
ad2antrr |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ v e. ( S ` suc N ) ) -> ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) ) |
60 |
|
eldifi |
|- ( u e. ( ( S ` suc N ) \ ( S ` N ) ) -> u e. ( S ` suc N ) ) |
61 |
60
|
adantl |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> u e. ( S ` suc N ) ) |
62 |
61
|
anim1i |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ v e. ( S ` suc N ) ) -> ( u e. ( S ` suc N ) /\ v e. ( S ` suc N ) ) ) |
63 |
55 59 62
|
3jca |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ v e. ( S ` suc N ) ) -> ( Fun ( S ` suc N ) /\ ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` suc N ) /\ v e. ( S ` suc N ) ) ) ) |
64 |
|
id |
|- ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) |
65 |
2
|
eqeq2i |
|- ( w = A <-> w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) |
66 |
65
|
biimpi |
|- ( w = A -> w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) |
67 |
66
|
anim2i |
|- ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) |
68 |
|
satffunlem |
|- ( ( ( Fun ( S ` suc N ) /\ ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` suc N ) /\ v e. ( S ` suc N ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) /\ ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) -> y = w ) |
69 |
63 64 67 68
|
syl3an |
|- ( ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ v e. ( S ` suc N ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) /\ ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) |
70 |
69
|
3exp |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ v e. ( S ` suc N ) ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> y = w ) ) ) |
71 |
70
|
com23 |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ v e. ( S ` suc N ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> y = w ) ) ) |
72 |
71
|
rexlimdva |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> y = w ) ) ) |
73 |
|
eqeq1 |
|- ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) -> ( x = A.g i ( 1st ` u ) <-> ( ( 1st ` s ) |g ( 1st ` r ) ) = A.g i ( 1st ` u ) ) ) |
74 |
|
fvex |
|- ( 1st ` s ) e. _V |
75 |
|
fvex |
|- ( 1st ` r ) e. _V |
76 |
|
gonafv |
|- ( ( ( 1st ` s ) e. _V /\ ( 1st ` r ) e. _V ) -> ( ( 1st ` s ) |g ( 1st ` r ) ) = <. 1o , <. ( 1st ` s ) , ( 1st ` r ) >. >. ) |
77 |
74 75 76
|
mp2an |
|- ( ( 1st ` s ) |g ( 1st ` r ) ) = <. 1o , <. ( 1st ` s ) , ( 1st ` r ) >. >. |
78 |
|
df-goal |
|- A.g i ( 1st ` u ) = <. 2o , <. i , ( 1st ` u ) >. >. |
79 |
77 78
|
eqeq12i |
|- ( ( ( 1st ` s ) |g ( 1st ` r ) ) = A.g i ( 1st ` u ) <-> <. 1o , <. ( 1st ` s ) , ( 1st ` r ) >. >. = <. 2o , <. i , ( 1st ` u ) >. >. ) |
80 |
|
1oex |
|- 1o e. _V |
81 |
|
opex |
|- <. ( 1st ` s ) , ( 1st ` r ) >. e. _V |
82 |
80 81
|
opth |
|- ( <. 1o , <. ( 1st ` s ) , ( 1st ` r ) >. >. = <. 2o , <. i , ( 1st ` u ) >. >. <-> ( 1o = 2o /\ <. ( 1st ` s ) , ( 1st ` r ) >. = <. i , ( 1st ` u ) >. ) ) |
83 |
|
1one2o |
|- 1o =/= 2o |
84 |
|
df-ne |
|- ( 1o =/= 2o <-> -. 1o = 2o ) |
85 |
|
pm2.21 |
|- ( -. 1o = 2o -> ( 1o = 2o -> y = w ) ) |
86 |
84 85
|
sylbi |
|- ( 1o =/= 2o -> ( 1o = 2o -> y = w ) ) |
87 |
83 86
|
ax-mp |
|- ( 1o = 2o -> y = w ) |
88 |
87
|
adantr |
|- ( ( 1o = 2o /\ <. ( 1st ` s ) , ( 1st ` r ) >. = <. i , ( 1st ` u ) >. ) -> y = w ) |
89 |
82 88
|
sylbi |
|- ( <. 1o , <. ( 1st ` s ) , ( 1st ` r ) >. >. = <. 2o , <. i , ( 1st ` u ) >. >. -> y = w ) |
90 |
79 89
|
sylbi |
|- ( ( ( 1st ` s ) |g ( 1st ` r ) ) = A.g i ( 1st ` u ) -> y = w ) |
91 |
73 90
|
syl6bi |
|- ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) -> ( x = A.g i ( 1st ` u ) -> y = w ) ) |
92 |
91
|
adantr |
|- ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( x = A.g i ( 1st ` u ) -> y = w ) ) |
93 |
92
|
com12 |
|- ( x = A.g i ( 1st ` u ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> y = w ) ) |
94 |
93
|
adantr |
|- ( ( x = A.g i ( 1st ` u ) /\ w = B ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> y = w ) ) |
95 |
94
|
a1i |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ i e. _om ) -> ( ( x = A.g i ( 1st ` u ) /\ w = B ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> y = w ) ) ) |
96 |
95
|
rexlimdva |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> y = w ) ) ) |
97 |
72 96
|
jaod |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> y = w ) ) ) |
98 |
97
|
rexlimdva |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) -> ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> y = w ) ) ) |
99 |
|
simp-4l |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> Fun ( S ` suc N ) ) |
100 |
58
|
adantr |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) ) |
101 |
|
ssel |
|- ( ( S ` N ) C_ ( S ` suc N ) -> ( u e. ( S ` N ) -> u e. ( S ` suc N ) ) ) |
102 |
101
|
ad3antlr |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) -> ( u e. ( S ` N ) -> u e. ( S ` suc N ) ) ) |
103 |
102
|
com12 |
|- ( u e. ( S ` N ) -> ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) -> u e. ( S ` suc N ) ) ) |
104 |
103
|
adantr |
|- ( ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) -> u e. ( S ` suc N ) ) ) |
105 |
104
|
impcom |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> u e. ( S ` suc N ) ) |
106 |
|
eldifi |
|- ( v e. ( ( S ` suc N ) \ ( S ` N ) ) -> v e. ( S ` suc N ) ) |
107 |
106
|
ad2antll |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> v e. ( S ` suc N ) ) |
108 |
105 107
|
jca |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( u e. ( S ` suc N ) /\ v e. ( S ` suc N ) ) ) |
109 |
99 100 108
|
3jca |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( Fun ( S ` suc N ) /\ ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` suc N ) /\ v e. ( S ` suc N ) ) ) ) |
110 |
109 64 67 68
|
syl3an |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) /\ ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) |
111 |
110
|
3exp |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> y = w ) ) ) |
112 |
111
|
com23 |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> y = w ) ) ) |
113 |
112
|
rexlimdvva |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) -> ( E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> y = w ) ) ) |
114 |
98 113
|
jaod |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> y = w ) ) ) |
115 |
114
|
com23 |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ r e. ( S ` suc N ) ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) ) ) |
116 |
115
|
rexlimdva |
|- ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( E. r e. ( S ` suc N ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) ) ) |
117 |
|
eqeq1 |
|- ( x = A.g j ( 1st ` s ) -> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) <-> A.g j ( 1st ` s ) = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
118 |
|
df-goal |
|- A.g j ( 1st ` s ) = <. 2o , <. j , ( 1st ` s ) >. >. |
119 |
|
fvex |
|- ( 1st ` u ) e. _V |
120 |
|
fvex |
|- ( 1st ` v ) e. _V |
121 |
|
gonafv |
|- ( ( ( 1st ` u ) e. _V /\ ( 1st ` v ) e. _V ) -> ( ( 1st ` u ) |g ( 1st ` v ) ) = <. 1o , <. ( 1st ` u ) , ( 1st ` v ) >. >. ) |
122 |
119 120 121
|
mp2an |
|- ( ( 1st ` u ) |g ( 1st ` v ) ) = <. 1o , <. ( 1st ` u ) , ( 1st ` v ) >. >. |
123 |
118 122
|
eqeq12i |
|- ( A.g j ( 1st ` s ) = ( ( 1st ` u ) |g ( 1st ` v ) ) <-> <. 2o , <. j , ( 1st ` s ) >. >. = <. 1o , <. ( 1st ` u ) , ( 1st ` v ) >. >. ) |
124 |
|
2oex |
|- 2o e. _V |
125 |
|
opex |
|- <. j , ( 1st ` s ) >. e. _V |
126 |
124 125
|
opth |
|- ( <. 2o , <. j , ( 1st ` s ) >. >. = <. 1o , <. ( 1st ` u ) , ( 1st ` v ) >. >. <-> ( 2o = 1o /\ <. j , ( 1st ` s ) >. = <. ( 1st ` u ) , ( 1st ` v ) >. ) ) |
127 |
87
|
eqcoms |
|- ( 2o = 1o -> y = w ) |
128 |
127
|
adantr |
|- ( ( 2o = 1o /\ <. j , ( 1st ` s ) >. = <. ( 1st ` u ) , ( 1st ` v ) >. ) -> y = w ) |
129 |
126 128
|
sylbi |
|- ( <. 2o , <. j , ( 1st ` s ) >. >. = <. 1o , <. ( 1st ` u ) , ( 1st ` v ) >. >. -> y = w ) |
130 |
123 129
|
sylbi |
|- ( A.g j ( 1st ` s ) = ( ( 1st ` u ) |g ( 1st ` v ) ) -> y = w ) |
131 |
117 130
|
syl6bi |
|- ( x = A.g j ( 1st ` s ) -> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) -> y = w ) ) |
132 |
131
|
adantr |
|- ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) -> y = w ) ) |
133 |
132
|
com12 |
|- ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) |
134 |
133
|
adantr |
|- ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) |
135 |
134
|
rexlimivw |
|- ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) |
136 |
135
|
a1i |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) ) |
137 |
|
eqeq1 |
|- ( x = A.g i ( 1st ` u ) -> ( x = A.g j ( 1st ` s ) <-> A.g i ( 1st ` u ) = A.g j ( 1st ` s ) ) ) |
138 |
78 118
|
eqeq12i |
|- ( A.g i ( 1st ` u ) = A.g j ( 1st ` s ) <-> <. 2o , <. i , ( 1st ` u ) >. >. = <. 2o , <. j , ( 1st ` s ) >. >. ) |
139 |
|
opex |
|- <. i , ( 1st ` u ) >. e. _V |
140 |
124 139
|
opth |
|- ( <. 2o , <. i , ( 1st ` u ) >. >. = <. 2o , <. j , ( 1st ` s ) >. >. <-> ( 2o = 2o /\ <. i , ( 1st ` u ) >. = <. j , ( 1st ` s ) >. ) ) |
141 |
|
vex |
|- i e. _V |
142 |
141 119
|
opth |
|- ( <. i , ( 1st ` u ) >. = <. j , ( 1st ` s ) >. <-> ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) ) |
143 |
142
|
anbi2i |
|- ( ( 2o = 2o /\ <. i , ( 1st ` u ) >. = <. j , ( 1st ` s ) >. ) <-> ( 2o = 2o /\ ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) ) ) |
144 |
138 140 143
|
3bitri |
|- ( A.g i ( 1st ` u ) = A.g j ( 1st ` s ) <-> ( 2o = 2o /\ ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) ) ) |
145 |
137 144
|
bitrdi |
|- ( x = A.g i ( 1st ` u ) -> ( x = A.g j ( 1st ` s ) <-> ( 2o = 2o /\ ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) ) ) ) |
146 |
145
|
adantl |
|- ( ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ i e. _om ) /\ x = A.g i ( 1st ` u ) ) -> ( x = A.g j ( 1st ` s ) <-> ( 2o = 2o /\ ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) ) ) ) |
147 |
56
|
a1i |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( s e. ( ( S ` suc N ) \ ( S ` N ) ) -> s e. ( S ` suc N ) ) ) |
148 |
|
funfv1st2nd |
|- ( ( Fun ( S ` suc N ) /\ s e. ( S ` suc N ) ) -> ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) ) |
149 |
148
|
ex |
|- ( Fun ( S ` suc N ) -> ( s e. ( S ` suc N ) -> ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) ) ) |
150 |
149
|
adantr |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( s e. ( S ` suc N ) -> ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) ) ) |
151 |
|
funfv1st2nd |
|- ( ( Fun ( S ` suc N ) /\ u e. ( S ` suc N ) ) -> ( ( S ` suc N ) ` ( 1st ` u ) ) = ( 2nd ` u ) ) |
152 |
151
|
ex |
|- ( Fun ( S ` suc N ) -> ( u e. ( S ` suc N ) -> ( ( S ` suc N ) ` ( 1st ` u ) ) = ( 2nd ` u ) ) ) |
153 |
152
|
adantr |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( u e. ( S ` suc N ) -> ( ( S ` suc N ) ` ( 1st ` u ) ) = ( 2nd ` u ) ) ) |
154 |
|
fveqeq2 |
|- ( ( 1st ` u ) = ( 1st ` s ) -> ( ( ( S ` suc N ) ` ( 1st ` u ) ) = ( 2nd ` u ) <-> ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` u ) ) ) |
155 |
|
eqtr2 |
|- ( ( ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` u ) /\ ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) ) -> ( 2nd ` u ) = ( 2nd ` s ) ) |
156 |
29
|
eqcomd |
|- ( i = j -> ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) = ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) ) |
157 |
156
|
adantl |
|- ( ( ( 2nd ` u ) = ( 2nd ` s ) /\ i = j ) -> ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) = ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) ) |
158 |
|
simpl |
|- ( ( ( 2nd ` u ) = ( 2nd ` s ) /\ i = j ) -> ( 2nd ` u ) = ( 2nd ` s ) ) |
159 |
158
|
eqcomd |
|- ( ( ( 2nd ` u ) = ( 2nd ` s ) /\ i = j ) -> ( 2nd ` s ) = ( 2nd ` u ) ) |
160 |
157 159
|
eleq12d |
|- ( ( ( 2nd ` u ) = ( 2nd ` s ) /\ i = j ) -> ( ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) <-> ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) ) ) |
161 |
160
|
ralbidv |
|- ( ( ( 2nd ` u ) = ( 2nd ` s ) /\ i = j ) -> ( A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) <-> A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) ) ) |
162 |
161
|
rabbidv |
|- ( ( ( 2nd ` u ) = ( 2nd ` s ) /\ i = j ) -> { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } = { a e. ( M ^m _om ) | A. z e. M ( { <. i , z >. } u. ( a |` ( _om \ { i } ) ) ) e. ( 2nd ` u ) } ) |
163 |
162 3
|
eqtr4di |
|- ( ( ( 2nd ` u ) = ( 2nd ` s ) /\ i = j ) -> { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } = B ) |
164 |
|
eqeq12 |
|- ( ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } /\ w = B ) -> ( y = w <-> { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } = B ) ) |
165 |
163 164
|
syl5ibrcom |
|- ( ( ( 2nd ` u ) = ( 2nd ` s ) /\ i = j ) -> ( ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } /\ w = B ) -> y = w ) ) |
166 |
165
|
exp4b |
|- ( ( 2nd ` u ) = ( 2nd ` s ) -> ( i = j -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) |
167 |
155 166
|
syl |
|- ( ( ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` u ) /\ ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) ) -> ( i = j -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) |
168 |
167
|
ex |
|- ( ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` u ) -> ( ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) -> ( i = j -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) ) |
169 |
154 168
|
syl6bi |
|- ( ( 1st ` u ) = ( 1st ` s ) -> ( ( ( S ` suc N ) ` ( 1st ` u ) ) = ( 2nd ` u ) -> ( ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) -> ( i = j -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) ) ) |
170 |
169
|
com24 |
|- ( ( 1st ` u ) = ( 1st ` s ) -> ( i = j -> ( ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) -> ( ( ( S ` suc N ) ` ( 1st ` u ) ) = ( 2nd ` u ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) ) ) |
171 |
170
|
impcom |
|- ( ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) -> ( ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) -> ( ( ( S ` suc N ) ` ( 1st ` u ) ) = ( 2nd ` u ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) ) |
172 |
171
|
com13 |
|- ( ( ( S ` suc N ) ` ( 1st ` u ) ) = ( 2nd ` u ) -> ( ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) -> ( ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) ) |
173 |
60 153 172
|
syl56 |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( u e. ( ( S ` suc N ) \ ( S ` N ) ) -> ( ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) -> ( ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) ) ) |
174 |
173
|
com23 |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( ( ( S ` suc N ) ` ( 1st ` s ) ) = ( 2nd ` s ) -> ( u e. ( ( S ` suc N ) \ ( S ` N ) ) -> ( ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) ) ) |
175 |
147 150 174
|
3syld |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( s e. ( ( S ` suc N ) \ ( S ` N ) ) -> ( u e. ( ( S ` suc N ) \ ( S ` N ) ) -> ( ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) ) ) |
176 |
175
|
imp |
|- ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( u e. ( ( S ` suc N ) \ ( S ` N ) ) -> ( ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) ) |
177 |
176
|
adantr |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) -> ( u e. ( ( S ` suc N ) \ ( S ` N ) ) -> ( ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) ) |
178 |
177
|
imp |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) |
179 |
178
|
adantld |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( 2o = 2o /\ ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) |
180 |
179
|
ad2antrr |
|- ( ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ i e. _om ) /\ x = A.g i ( 1st ` u ) ) -> ( ( 2o = 2o /\ ( i = j /\ ( 1st ` u ) = ( 1st ` s ) ) ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) |
181 |
146 180
|
sylbid |
|- ( ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ i e. _om ) /\ x = A.g i ( 1st ` u ) ) -> ( x = A.g j ( 1st ` s ) -> ( y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } -> ( w = B -> y = w ) ) ) ) |
182 |
181
|
impd |
|- ( ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ i e. _om ) /\ x = A.g i ( 1st ` u ) ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> ( w = B -> y = w ) ) ) |
183 |
182
|
ex |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ i e. _om ) -> ( x = A.g i ( 1st ` u ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> ( w = B -> y = w ) ) ) ) |
184 |
183
|
com34 |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ i e. _om ) -> ( x = A.g i ( 1st ` u ) -> ( w = B -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) ) ) |
185 |
184
|
impd |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ i e. _om ) -> ( ( x = A.g i ( 1st ` u ) /\ w = B ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) ) |
186 |
185
|
rexlimdva |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) ) |
187 |
136 186
|
jaod |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) ) |
188 |
187
|
rexlimdva |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) -> ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) ) |
189 |
134
|
a1i |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( S ` N ) ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) ) |
190 |
189
|
rexlimdva |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) /\ u e. ( S ` N ) ) -> ( E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) ) |
191 |
190
|
rexlimdva |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) -> ( E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) ) |
192 |
188 191
|
jaod |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> y = w ) ) ) |
193 |
192
|
com23 |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ j e. _om ) -> ( ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) ) ) |
194 |
193
|
rexlimdva |
|- ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( E. j e. _om ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) ) ) |
195 |
116 194
|
jaod |
|- ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ s e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( E. r e. ( S ` suc N ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) \/ E. j e. _om ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) ) ) |
196 |
195
|
rexlimdva |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( E. s e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. r e. ( S ` suc N ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) \/ E. j e. _om ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) ) ) |
197 |
|
simplll |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( ( S ` suc N ) \ ( S ` N ) ) /\ v e. ( S ` suc N ) ) ) -> Fun ( S ` suc N ) ) |
198 |
|
ssel |
|- ( ( S ` N ) C_ ( S ` suc N ) -> ( s e. ( S ` N ) -> s e. ( S ` suc N ) ) ) |
199 |
198
|
adantrd |
|- ( ( S ` N ) C_ ( S ` suc N ) -> ( ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> s e. ( S ` suc N ) ) ) |
200 |
199
|
adantl |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> s e. ( S ` suc N ) ) ) |
201 |
200
|
imp |
|- ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> s e. ( S ` suc N ) ) |
202 |
|
eldifi |
|- ( r e. ( ( S ` suc N ) \ ( S ` N ) ) -> r e. ( S ` suc N ) ) |
203 |
202
|
ad2antll |
|- ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> r e. ( S ` suc N ) ) |
204 |
201 203
|
jca |
|- ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) ) |
205 |
204
|
adantr |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( ( S ` suc N ) \ ( S ` N ) ) /\ v e. ( S ` suc N ) ) ) -> ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) ) |
206 |
60
|
anim1i |
|- ( ( u e. ( ( S ` suc N ) \ ( S ` N ) ) /\ v e. ( S ` suc N ) ) -> ( u e. ( S ` suc N ) /\ v e. ( S ` suc N ) ) ) |
207 |
206
|
adantl |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( ( S ` suc N ) \ ( S ` N ) ) /\ v e. ( S ` suc N ) ) ) -> ( u e. ( S ` suc N ) /\ v e. ( S ` suc N ) ) ) |
208 |
197 205 207
|
3jca |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( ( S ` suc N ) \ ( S ` N ) ) /\ v e. ( S ` suc N ) ) ) -> ( Fun ( S ` suc N ) /\ ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` suc N ) /\ v e. ( S ` suc N ) ) ) ) |
209 |
208
|
adantr |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( ( S ` suc N ) \ ( S ` N ) ) /\ v e. ( S ` suc N ) ) ) /\ ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) /\ ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) ) -> ( Fun ( S ` suc N ) /\ ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` suc N ) /\ v e. ( S ` suc N ) ) ) ) |
210 |
|
simprl |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( ( S ` suc N ) \ ( S ` N ) ) /\ v e. ( S ` suc N ) ) ) /\ ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) /\ ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) ) -> ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) |
211 |
67
|
ad2antll |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( ( S ` suc N ) \ ( S ` N ) ) /\ v e. ( S ` suc N ) ) ) /\ ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) /\ ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) ) -> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = ( ( M ^m _om ) \ ( ( 2nd ` u ) i^i ( 2nd ` v ) ) ) ) ) |
212 |
209 210 211 68
|
syl3anc |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( ( S ` suc N ) \ ( S ` N ) ) /\ v e. ( S ` suc N ) ) ) /\ ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) /\ ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) ) -> y = w ) |
213 |
212
|
exp32 |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( ( S ` suc N ) \ ( S ` N ) ) /\ v e. ( S ` suc N ) ) ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> y = w ) ) ) |
214 |
213
|
impancom |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) -> ( ( u e. ( ( S ` suc N ) \ ( S ` N ) ) /\ v e. ( S ` suc N ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> y = w ) ) ) |
215 |
214
|
expdimp |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( v e. ( S ` suc N ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> y = w ) ) ) |
216 |
215
|
rexlimdv |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> y = w ) ) |
217 |
91
|
adantrd |
|- ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) -> ( ( x = A.g i ( 1st ` u ) /\ w = B ) -> y = w ) ) |
218 |
217
|
adantr |
|- ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( ( x = A.g i ( 1st ` u ) /\ w = B ) -> y = w ) ) |
219 |
218
|
ad3antlr |
|- ( ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) /\ i e. _om ) -> ( ( x = A.g i ( 1st ` u ) /\ w = B ) -> y = w ) ) |
220 |
219
|
rexlimdva |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) -> y = w ) ) |
221 |
216 220
|
jaod |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) /\ u e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) -> y = w ) ) |
222 |
221
|
rexlimdva |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) -> ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) -> y = w ) ) |
223 |
|
simplll |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> Fun ( S ` suc N ) ) |
224 |
204
|
adantr |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) ) |
225 |
101
|
adantl |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( u e. ( S ` N ) -> u e. ( S ` suc N ) ) ) |
226 |
225
|
adantr |
|- ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( u e. ( S ` N ) -> u e. ( S ` suc N ) ) ) |
227 |
226
|
com12 |
|- ( u e. ( S ` N ) -> ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> u e. ( S ` suc N ) ) ) |
228 |
227
|
adantr |
|- ( ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> u e. ( S ` suc N ) ) ) |
229 |
228
|
impcom |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> u e. ( S ` suc N ) ) |
230 |
106
|
ad2antll |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> v e. ( S ` suc N ) ) |
231 |
229 230
|
jca |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( u e. ( S ` suc N ) /\ v e. ( S ` suc N ) ) ) |
232 |
223 224 231
|
3jca |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( Fun ( S ` suc N ) /\ ( s e. ( S ` suc N ) /\ r e. ( S ` suc N ) ) /\ ( u e. ( S ` suc N ) /\ v e. ( S ` suc N ) ) ) ) |
233 |
232 64 67 68
|
syl3an |
|- ( ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) /\ ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) |
234 |
233
|
3exp |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> y = w ) ) ) |
235 |
234
|
impancom |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) -> ( ( u e. ( S ` N ) /\ v e. ( ( S ` suc N ) \ ( S ` N ) ) ) -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> y = w ) ) ) |
236 |
235
|
rexlimdvv |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) -> ( E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) -> y = w ) ) |
237 |
222 236
|
jaod |
|- ( ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) /\ ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) ) |
238 |
237
|
ex |
|- ( ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) /\ ( s e. ( S ` N ) /\ r e. ( ( S ` suc N ) \ ( S ` N ) ) ) ) -> ( ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) ) ) |
239 |
238
|
rexlimdvva |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( E. s e. ( S ` N ) E. r e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) ) ) |
240 |
196 239
|
jaod |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( ( E. s e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. r e. ( S ` suc N ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) \/ E. j e. _om ( x = A.g j ( 1st ` s ) /\ y = { a e. ( M ^m _om ) | A. z e. M ( { <. j , z >. } u. ( a |` ( _om \ { j } ) ) ) e. ( 2nd ` s ) } ) ) \/ E. s e. ( S ` N ) E. r e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` s ) |g ( 1st ` r ) ) /\ y = ( ( M ^m _om ) \ ( ( 2nd ` s ) i^i ( 2nd ` r ) ) ) ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) ) ) |
241 |
54 240
|
syl5bi |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( ( 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 ) ) -> ( ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) -> y = w ) ) ) |
242 |
241
|
impd |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> ( ( ( 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 ) ) /\ ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) ) -> y = w ) ) |
243 |
242
|
alrimivv |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> A. y A. w ( ( ( 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 ) ) /\ ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) ) -> y = w ) ) |
244 |
|
eqeq1 |
|- ( y = w -> ( y = A <-> w = A ) ) |
245 |
244
|
anbi2d |
|- ( y = w -> ( ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) <-> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) ) |
246 |
245
|
rexbidv |
|- ( y = w -> ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) <-> E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) ) |
247 |
|
eqeq1 |
|- ( y = w -> ( y = B <-> w = B ) ) |
248 |
247
|
anbi2d |
|- ( y = w -> ( ( x = A.g i ( 1st ` u ) /\ y = B ) <-> ( x = A.g i ( 1st ` u ) /\ w = B ) ) ) |
249 |
248
|
rexbidv |
|- ( y = w -> ( E. i e. _om ( x = A.g i ( 1st ` u ) /\ y = B ) <-> E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) ) |
250 |
246 249
|
orbi12d |
|- ( y = w -> ( ( 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. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) ) ) |
251 |
250
|
rexbidv |
|- ( y = w -> ( 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 ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) ) ) |
252 |
245
|
2rexbidv |
|- ( y = w -> ( E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ y = A ) <-> E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) ) |
253 |
251 252
|
orbi12d |
|- ( y = w -> ( ( 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 ) ) <-> ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) ) ) |
254 |
253
|
mo4 |
|- ( E* 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 ) ) <-> A. y A. w ( ( ( 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 ) ) /\ ( E. u e. ( ( S ` suc N ) \ ( S ` N ) ) ( E. v e. ( S ` suc N ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) \/ E. i e. _om ( x = A.g i ( 1st ` u ) /\ w = B ) ) \/ E. u e. ( S ` N ) E. v e. ( ( S ` suc N ) \ ( S ` N ) ) ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) /\ w = A ) ) ) -> y = w ) ) |
255 |
243 254
|
sylibr |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> E* 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 ) ) ) |
256 |
255
|
alrimiv |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> A. x E* 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 ) ) ) |
257 |
|
funopab |
|- ( Fun { <. 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 ) ) } <-> A. x E* 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 ) ) ) |
258 |
256 257
|
sylibr |
|- ( ( Fun ( S ` suc N ) /\ ( S ` N ) C_ ( S ` suc N ) ) -> Fun { <. 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 ) ) } ) |