Step |
Hyp |
Ref |
Expression |
1 |
|
satf0op.s |
|- S = ( (/) Sat (/) ) |
2 |
|
fveq2 |
|- ( y = (/) -> ( S ` y ) = ( S ` (/) ) ) |
3 |
2
|
eleq2d |
|- ( y = (/) -> ( X e. ( S ` y ) <-> X e. ( S ` (/) ) ) ) |
4 |
2
|
eleq2d |
|- ( y = (/) -> ( <. x , (/) >. e. ( S ` y ) <-> <. x , (/) >. e. ( S ` (/) ) ) ) |
5 |
4
|
anbi2d |
|- ( y = (/) -> ( ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) <-> ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` (/) ) ) ) ) |
6 |
5
|
exbidv |
|- ( y = (/) -> ( E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` (/) ) ) ) ) |
7 |
3 6
|
bibi12d |
|- ( y = (/) -> ( ( X e. ( S ` y ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) ) <-> ( X e. ( S ` (/) ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` (/) ) ) ) ) ) |
8 |
|
fveq2 |
|- ( y = z -> ( S ` y ) = ( S ` z ) ) |
9 |
8
|
eleq2d |
|- ( y = z -> ( X e. ( S ` y ) <-> X e. ( S ` z ) ) ) |
10 |
8
|
eleq2d |
|- ( y = z -> ( <. x , (/) >. e. ( S ` y ) <-> <. x , (/) >. e. ( S ` z ) ) ) |
11 |
10
|
anbi2d |
|- ( y = z -> ( ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) <-> ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) ) |
12 |
11
|
exbidv |
|- ( y = z -> ( E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) ) |
13 |
9 12
|
bibi12d |
|- ( y = z -> ( ( X e. ( S ` y ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) ) <-> ( X e. ( S ` z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) ) ) |
14 |
|
fveq2 |
|- ( y = suc z -> ( S ` y ) = ( S ` suc z ) ) |
15 |
14
|
eleq2d |
|- ( y = suc z -> ( X e. ( S ` y ) <-> X e. ( S ` suc z ) ) ) |
16 |
14
|
eleq2d |
|- ( y = suc z -> ( <. x , (/) >. e. ( S ` y ) <-> <. x , (/) >. e. ( S ` suc z ) ) ) |
17 |
16
|
anbi2d |
|- ( y = suc z -> ( ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) <-> ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` suc z ) ) ) ) |
18 |
17
|
exbidv |
|- ( y = suc z -> ( E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` suc z ) ) ) ) |
19 |
15 18
|
bibi12d |
|- ( y = suc z -> ( ( X e. ( S ` y ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) ) <-> ( X e. ( S ` suc z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` suc z ) ) ) ) ) |
20 |
|
fveq2 |
|- ( y = N -> ( S ` y ) = ( S ` N ) ) |
21 |
20
|
eleq2d |
|- ( y = N -> ( X e. ( S ` y ) <-> X e. ( S ` N ) ) ) |
22 |
20
|
eleq2d |
|- ( y = N -> ( <. x , (/) >. e. ( S ` y ) <-> <. x , (/) >. e. ( S ` N ) ) ) |
23 |
22
|
anbi2d |
|- ( y = N -> ( ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) <-> ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` N ) ) ) ) |
24 |
23
|
exbidv |
|- ( y = N -> ( E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` N ) ) ) ) |
25 |
21 24
|
bibi12d |
|- ( y = N -> ( ( X e. ( S ` y ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` y ) ) ) <-> ( X e. ( S ` N ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` N ) ) ) ) ) |
26 |
1
|
fveq1i |
|- ( S ` (/) ) = ( ( (/) Sat (/) ) ` (/) ) |
27 |
|
satf00 |
|- ( ( (/) Sat (/) ) ` (/) ) = { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } |
28 |
26 27
|
eqtri |
|- ( S ` (/) ) = { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } |
29 |
28
|
eleq2i |
|- ( X e. ( S ` (/) ) <-> X e. { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } ) |
30 |
|
elopab |
|- ( X e. { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } <-> E. x E. y ( X = <. x , y >. /\ ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) ) |
31 |
|
opeq2 |
|- ( y = (/) -> <. x , y >. = <. x , (/) >. ) |
32 |
31
|
adantr |
|- ( ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) -> <. x , y >. = <. x , (/) >. ) |
33 |
32
|
eqeq2d |
|- ( ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) -> ( X = <. x , y >. <-> X = <. x , (/) >. ) ) |
34 |
33
|
biimpd |
|- ( ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) -> ( X = <. x , y >. -> X = <. x , (/) >. ) ) |
35 |
34
|
impcom |
|- ( ( X = <. x , y >. /\ ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) -> X = <. x , (/) >. ) |
36 |
|
eqidd |
|- ( y = (/) -> (/) = (/) ) |
37 |
36
|
anim1i |
|- ( ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) -> ( (/) = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) |
38 |
37
|
adantl |
|- ( ( X = <. x , y >. /\ ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) -> ( (/) = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) |
39 |
|
satf00 |
|- ( ( (/) Sat (/) ) ` (/) ) = { <. y , z >. | ( z = (/) /\ E. i e. _om E. j e. _om y = ( i e.g j ) ) } |
40 |
26 39
|
eqtri |
|- ( S ` (/) ) = { <. y , z >. | ( z = (/) /\ E. i e. _om E. j e. _om y = ( i e.g j ) ) } |
41 |
40
|
eleq2i |
|- ( <. x , (/) >. e. ( S ` (/) ) <-> <. x , (/) >. e. { <. y , z >. | ( z = (/) /\ E. i e. _om E. j e. _om y = ( i e.g j ) ) } ) |
42 |
|
vex |
|- x e. _V |
43 |
|
0ex |
|- (/) e. _V |
44 |
|
eqeq1 |
|- ( z = (/) -> ( z = (/) <-> (/) = (/) ) ) |
45 |
|
eqeq1 |
|- ( y = x -> ( y = ( i e.g j ) <-> x = ( i e.g j ) ) ) |
46 |
45
|
2rexbidv |
|- ( y = x -> ( E. i e. _om E. j e. _om y = ( i e.g j ) <-> E. i e. _om E. j e. _om x = ( i e.g j ) ) ) |
47 |
44 46
|
bi2anan9r |
|- ( ( y = x /\ z = (/) ) -> ( ( z = (/) /\ E. i e. _om E. j e. _om y = ( i e.g j ) ) <-> ( (/) = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) ) |
48 |
42 43 47
|
opelopaba |
|- ( <. x , (/) >. e. { <. y , z >. | ( z = (/) /\ E. i e. _om E. j e. _om y = ( i e.g j ) ) } <-> ( (/) = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) |
49 |
41 48
|
bitri |
|- ( <. x , (/) >. e. ( S ` (/) ) <-> ( (/) = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) |
50 |
38 49
|
sylibr |
|- ( ( X = <. x , y >. /\ ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) -> <. x , (/) >. e. ( S ` (/) ) ) |
51 |
35 50
|
jca |
|- ( ( X = <. x , y >. /\ ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) -> ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` (/) ) ) ) |
52 |
51
|
exlimiv |
|- ( E. y ( X = <. x , y >. /\ ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) -> ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` (/) ) ) ) |
53 |
31
|
eqeq2d |
|- ( y = (/) -> ( X = <. x , y >. <-> X = <. x , (/) >. ) ) |
54 |
|
eqeq1 |
|- ( y = (/) -> ( y = (/) <-> (/) = (/) ) ) |
55 |
54
|
anbi1d |
|- ( y = (/) -> ( ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) <-> ( (/) = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) ) |
56 |
53 55
|
anbi12d |
|- ( y = (/) -> ( ( X = <. x , y >. /\ ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) <-> ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) ) ) |
57 |
43 56
|
spcev |
|- ( ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) -> E. y ( X = <. x , y >. /\ ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) ) |
58 |
49 57
|
sylan2b |
|- ( ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` (/) ) ) -> E. y ( X = <. x , y >. /\ ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) ) |
59 |
52 58
|
impbii |
|- ( E. y ( X = <. x , y >. /\ ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) <-> ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` (/) ) ) ) |
60 |
59
|
exbii |
|- ( E. x E. y ( X = <. x , y >. /\ ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` (/) ) ) ) |
61 |
29 30 60
|
3bitri |
|- ( X e. ( S ` (/) ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` (/) ) ) ) |
62 |
1
|
satf0suc |
|- ( z e. _om -> ( S ` suc z ) = ( ( S ` z ) u. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) ) |
63 |
62
|
eleq2d |
|- ( z e. _om -> ( X e. ( S ` suc z ) <-> X e. ( ( S ` z ) u. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) ) ) |
64 |
|
elun |
|- ( X e. ( ( S ` z ) u. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) <-> ( X e. ( S ` z ) \/ X e. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) ) |
65 |
64
|
a1i |
|- ( z e. _om -> ( X e. ( ( S ` z ) u. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) <-> ( X e. ( S ` z ) \/ X e. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) ) ) |
66 |
|
elopab |
|- ( X e. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } <-> E. a E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) |
67 |
66
|
a1i |
|- ( z e. _om -> ( X e. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } <-> E. a E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) ) |
68 |
67
|
orbi2d |
|- ( z e. _om -> ( ( X e. ( S ` z ) \/ X e. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) <-> ( X e. ( S ` z ) \/ E. a E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) ) ) |
69 |
63 65 68
|
3bitrd |
|- ( z e. _om -> ( X e. ( S ` suc z ) <-> ( X e. ( S ` z ) \/ E. a E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) ) ) |
70 |
69
|
adantr |
|- ( ( z e. _om /\ ( X e. ( S ` z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) ) -> ( X e. ( S ` suc z ) <-> ( X e. ( S ` z ) \/ E. a E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) ) ) |
71 |
|
simpr |
|- ( ( z e. _om /\ ( X e. ( S ` z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) ) -> ( X e. ( S ` z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) ) |
72 |
|
opeq2 |
|- ( b = (/) -> <. a , b >. = <. a , (/) >. ) |
73 |
72
|
eqeq2d |
|- ( b = (/) -> ( X = <. a , b >. <-> X = <. a , (/) >. ) ) |
74 |
73
|
biimpd |
|- ( b = (/) -> ( X = <. a , b >. -> X = <. a , (/) >. ) ) |
75 |
74
|
adantr |
|- ( ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) -> ( X = <. a , b >. -> X = <. a , (/) >. ) ) |
76 |
75
|
impcom |
|- ( ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) -> X = <. a , (/) >. ) |
77 |
|
eqidd |
|- ( ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) -> (/) = (/) ) |
78 |
|
simpr |
|- ( ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) -> E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) |
79 |
78
|
adantl |
|- ( ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) -> E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) |
80 |
77 79
|
jca |
|- ( ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) -> ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) |
81 |
76 80
|
jca |
|- ( ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) -> ( X = <. a , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) |
82 |
81
|
exlimiv |
|- ( E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) -> ( X = <. a , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) |
83 |
|
eqeq1 |
|- ( b = (/) -> ( b = (/) <-> (/) = (/) ) ) |
84 |
83
|
anbi1d |
|- ( b = (/) -> ( ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) <-> ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) |
85 |
73 84
|
anbi12d |
|- ( b = (/) -> ( ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) <-> ( X = <. a , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) ) |
86 |
43 85
|
spcev |
|- ( ( X = <. a , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) -> E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) |
87 |
82 86
|
impbii |
|- ( E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) <-> ( X = <. a , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) |
88 |
87
|
exbii |
|- ( E. a E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) <-> E. a ( X = <. a , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) |
89 |
88
|
a1i |
|- ( z e. _om -> ( E. a E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) <-> E. a ( X = <. a , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) ) |
90 |
|
opeq1 |
|- ( x = a -> <. x , (/) >. = <. a , (/) >. ) |
91 |
90
|
eqeq2d |
|- ( x = a -> ( X = <. x , (/) >. <-> X = <. a , (/) >. ) ) |
92 |
|
eqeq1 |
|- ( x = a -> ( x = ( ( 1st ` u ) |g ( 1st ` v ) ) <-> a = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
93 |
92
|
rexbidv |
|- ( x = a -> ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) <-> E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
94 |
|
eqeq1 |
|- ( x = a -> ( x = A.g i ( 1st ` u ) <-> a = A.g i ( 1st ` u ) ) ) |
95 |
94
|
rexbidv |
|- ( x = a -> ( E. i e. _om x = A.g i ( 1st ` u ) <-> E. i e. _om a = A.g i ( 1st ` u ) ) ) |
96 |
93 95
|
orbi12d |
|- ( x = a -> ( ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) <-> ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) |
97 |
96
|
rexbidv |
|- ( x = a -> ( E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) <-> E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) |
98 |
97
|
anbi2d |
|- ( x = a -> ( ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) <-> ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) |
99 |
91 98
|
anbi12d |
|- ( x = a -> ( ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) <-> ( X = <. a , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) ) |
100 |
99
|
cbvexvw |
|- ( E. x ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) <-> E. a ( X = <. a , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) |
101 |
89 100
|
bitr4di |
|- ( z e. _om -> ( E. a E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) <-> E. x ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) |
102 |
101
|
adantr |
|- ( ( z e. _om /\ ( X e. ( S ` z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) ) -> ( E. a E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) <-> E. x ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) |
103 |
71 102
|
orbi12d |
|- ( ( z e. _om /\ ( X e. ( S ` z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) ) -> ( ( X e. ( S ` z ) \/ E. a E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) <-> ( E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) \/ E. x ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) ) |
104 |
|
19.43 |
|- ( E. x ( ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) \/ ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) <-> ( E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) \/ E. x ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) |
105 |
|
andi |
|- ( ( X = <. x , (/) >. /\ ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) <-> ( ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) \/ ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) |
106 |
105
|
bicomi |
|- ( ( ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) \/ ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) <-> ( X = <. x , (/) >. /\ ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) |
107 |
106
|
exbii |
|- ( E. x ( ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) \/ ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) <-> E. x ( X = <. x , (/) >. /\ ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) |
108 |
104 107
|
bitr3i |
|- ( ( E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) \/ E. x ( X = <. x , (/) >. /\ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) <-> E. x ( X = <. x , (/) >. /\ ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) |
109 |
103 108
|
bitrdi |
|- ( ( z e. _om /\ ( X e. ( S ` z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) ) -> ( ( X e. ( S ` z ) \/ E. a E. b ( X = <. a , b >. /\ ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) ) ) <-> E. x ( X = <. x , (/) >. /\ ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) ) |
110 |
62
|
eleq2d |
|- ( z e. _om -> ( <. x , (/) >. e. ( S ` suc z ) <-> <. x , (/) >. e. ( ( S ` z ) u. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) ) ) |
111 |
|
elun |
|- ( <. x , (/) >. e. ( ( S ` z ) u. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) <-> ( <. x , (/) >. e. ( S ` z ) \/ <. x , (/) >. e. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) ) |
112 |
|
eqeq1 |
|- ( a = x -> ( a = ( ( 1st ` u ) |g ( 1st ` v ) ) <-> x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
113 |
112
|
rexbidv |
|- ( a = x -> ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) <-> E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) ) ) |
114 |
|
eqeq1 |
|- ( a = x -> ( a = A.g i ( 1st ` u ) <-> x = A.g i ( 1st ` u ) ) ) |
115 |
114
|
rexbidv |
|- ( a = x -> ( E. i e. _om a = A.g i ( 1st ` u ) <-> E. i e. _om x = A.g i ( 1st ` u ) ) ) |
116 |
113 115
|
orbi12d |
|- ( a = x -> ( ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) <-> ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
117 |
116
|
rexbidv |
|- ( a = x -> ( E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) <-> E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
118 |
83 117
|
bi2anan9r |
|- ( ( a = x /\ b = (/) ) -> ( ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) <-> ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) |
119 |
42 43 118
|
opelopaba |
|- ( <. x , (/) >. e. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } <-> ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) |
120 |
119
|
orbi2i |
|- ( ( <. x , (/) >. e. ( S ` z ) \/ <. x , (/) >. e. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) <-> ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) |
121 |
111 120
|
bitri |
|- ( <. x , (/) >. e. ( ( S ` z ) u. { <. a , b >. | ( b = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) a = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om a = A.g i ( 1st ` u ) ) ) } ) <-> ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) |
122 |
110 121
|
bitrdi |
|- ( z e. _om -> ( <. x , (/) >. e. ( S ` suc z ) <-> ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) |
123 |
122
|
anbi2d |
|- ( z e. _om -> ( ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` suc z ) ) <-> ( X = <. x , (/) >. /\ ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) ) |
124 |
123
|
exbidv |
|- ( z e. _om -> ( E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` suc z ) ) <-> E. x ( X = <. x , (/) >. /\ ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) ) ) |
125 |
124
|
bicomd |
|- ( z e. _om -> ( E. x ( X = <. x , (/) >. /\ ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` suc z ) ) ) ) |
126 |
125
|
adantr |
|- ( ( z e. _om /\ ( X e. ( S ` z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) ) -> ( E. x ( X = <. x , (/) >. /\ ( <. x , (/) >. e. ( S ` z ) \/ ( (/) = (/) /\ E. u e. ( S ` z ) ( E. v e. ( S ` z ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) ) ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` suc z ) ) ) ) |
127 |
70 109 126
|
3bitrd |
|- ( ( z e. _om /\ ( X e. ( S ` z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) ) -> ( X e. ( S ` suc z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` suc z ) ) ) ) |
128 |
127
|
ex |
|- ( z e. _om -> ( ( X e. ( S ` z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` z ) ) ) -> ( X e. ( S ` suc z ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` suc z ) ) ) ) ) |
129 |
7 13 19 25 61 128
|
finds |
|- ( N e. _om -> ( X e. ( S ` N ) <-> E. x ( X = <. x , (/) >. /\ <. x , (/) >. e. ( S ` N ) ) ) ) |