Step |
Hyp |
Ref |
Expression |
1 |
|
satefv |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( M SatE X ) = ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` X ) ) |
2 |
|
incom |
|- ( _E i^i ( M X. M ) ) = ( ( M X. M ) i^i _E ) |
3 |
|
sqxpexg |
|- ( M e. V -> ( M X. M ) e. _V ) |
4 |
|
inex1g |
|- ( ( M X. M ) e. _V -> ( ( M X. M ) i^i _E ) e. _V ) |
5 |
3 4
|
syl |
|- ( M e. V -> ( ( M X. M ) i^i _E ) e. _V ) |
6 |
2 5
|
eqeltrid |
|- ( M e. V -> ( _E i^i ( M X. M ) ) e. _V ) |
7 |
6
|
ancli |
|- ( M e. V -> ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V ) ) |
8 |
7
|
adantr |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V ) ) |
9 |
|
satom |
|- ( ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V ) -> ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) = U_ i e. _om ( ( M Sat ( _E i^i ( M X. M ) ) ) ` i ) ) |
10 |
8 9
|
syl |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) = U_ i e. _om ( ( M Sat ( _E i^i ( M X. M ) ) ) ` i ) ) |
11 |
10
|
fveq1d |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` X ) = ( U_ i e. _om ( ( M Sat ( _E i^i ( M X. M ) ) ) ` i ) ` X ) ) |
12 |
|
satfun |
|- ( ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V ) -> ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) : ( Fmla ` _om ) --> ~P ( M ^m _om ) ) |
13 |
8 12
|
syl |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) : ( Fmla ` _om ) --> ~P ( M ^m _om ) ) |
14 |
13
|
ffund |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> Fun ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ) |
15 |
10
|
eqcomd |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> U_ i e. _om ( ( M Sat ( _E i^i ( M X. M ) ) ) ` i ) = ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ) |
16 |
15
|
funeqd |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( Fun U_ i e. _om ( ( M Sat ( _E i^i ( M X. M ) ) ) ` i ) <-> Fun ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ) ) |
17 |
14 16
|
mpbird |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> Fun U_ i e. _om ( ( M Sat ( _E i^i ( M X. M ) ) ) ` i ) ) |
18 |
|
peano1 |
|- (/) e. _om |
19 |
18
|
a1i |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> (/) e. _om ) |
20 |
18
|
a1i |
|- ( M e. V -> (/) e. _om ) |
21 |
|
satfdmfmla |
|- ( ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V /\ (/) e. _om ) -> dom ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) = ( Fmla ` (/) ) ) |
22 |
6 20 21
|
mpd3an23 |
|- ( M e. V -> dom ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) = ( Fmla ` (/) ) ) |
23 |
22
|
eqcomd |
|- ( M e. V -> ( Fmla ` (/) ) = dom ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) ) |
24 |
23
|
eleq2d |
|- ( M e. V -> ( X e. ( Fmla ` (/) ) <-> X e. dom ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) ) ) |
25 |
24
|
biimpa |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> X e. dom ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) ) |
26 |
|
eqid |
|- U_ i e. _om ( ( M Sat ( _E i^i ( M X. M ) ) ) ` i ) = U_ i e. _om ( ( M Sat ( _E i^i ( M X. M ) ) ) ` i ) |
27 |
26
|
fviunfun |
|- ( ( Fun U_ i e. _om ( ( M Sat ( _E i^i ( M X. M ) ) ) ` i ) /\ (/) e. _om /\ X e. dom ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) ) -> ( U_ i e. _om ( ( M Sat ( _E i^i ( M X. M ) ) ) ` i ) ` X ) = ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) ` X ) ) |
28 |
17 19 25 27
|
syl3anc |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( U_ i e. _om ( ( M Sat ( _E i^i ( M X. M ) ) ) ` i ) ` X ) = ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) ` X ) ) |
29 |
11 28
|
eqtrd |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` X ) = ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) ` X ) ) |
30 |
|
simpl |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> M e. V ) |
31 |
6
|
adantr |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( _E i^i ( M X. M ) ) e. _V ) |
32 |
|
simpr |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> X e. ( Fmla ` (/) ) ) |
33 |
|
eqid |
|- ( M Sat ( _E i^i ( M X. M ) ) ) = ( M Sat ( _E i^i ( M X. M ) ) ) |
34 |
33
|
satfv0fvfmla0 |
|- ( ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V /\ X e. ( Fmla ` (/) ) ) -> ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) ` X ) = { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` X ) ) ) ( _E i^i ( M X. M ) ) ( a ` ( 2nd ` ( 2nd ` X ) ) ) } ) |
35 |
30 31 32 34
|
syl3anc |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) ` X ) = { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` X ) ) ) ( _E i^i ( M X. M ) ) ( a ` ( 2nd ` ( 2nd ` X ) ) ) } ) |
36 |
|
elmapi |
|- ( a e. ( M ^m _om ) -> a : _om --> M ) |
37 |
|
simpl |
|- ( ( a : _om --> M /\ ( M e. V /\ X e. ( Fmla ` (/) ) ) ) -> a : _om --> M ) |
38 |
|
fmla0xp |
|- ( Fmla ` (/) ) = ( { (/) } X. ( _om X. _om ) ) |
39 |
38
|
eleq2i |
|- ( X e. ( Fmla ` (/) ) <-> X e. ( { (/) } X. ( _om X. _om ) ) ) |
40 |
|
elxp |
|- ( X e. ( { (/) } X. ( _om X. _om ) ) <-> E. x E. y ( X = <. x , y >. /\ ( x e. { (/) } /\ y e. ( _om X. _om ) ) ) ) |
41 |
39 40
|
bitri |
|- ( X e. ( Fmla ` (/) ) <-> E. x E. y ( X = <. x , y >. /\ ( x e. { (/) } /\ y e. ( _om X. _om ) ) ) ) |
42 |
|
xp1st |
|- ( y e. ( _om X. _om ) -> ( 1st ` y ) e. _om ) |
43 |
42
|
ad2antll |
|- ( ( X = <. x , y >. /\ ( x e. { (/) } /\ y e. ( _om X. _om ) ) ) -> ( 1st ` y ) e. _om ) |
44 |
|
vex |
|- x e. _V |
45 |
|
vex |
|- y e. _V |
46 |
44 45
|
op2ndd |
|- ( X = <. x , y >. -> ( 2nd ` X ) = y ) |
47 |
46
|
fveq2d |
|- ( X = <. x , y >. -> ( 1st ` ( 2nd ` X ) ) = ( 1st ` y ) ) |
48 |
47
|
eleq1d |
|- ( X = <. x , y >. -> ( ( 1st ` ( 2nd ` X ) ) e. _om <-> ( 1st ` y ) e. _om ) ) |
49 |
48
|
adantr |
|- ( ( X = <. x , y >. /\ ( x e. { (/) } /\ y e. ( _om X. _om ) ) ) -> ( ( 1st ` ( 2nd ` X ) ) e. _om <-> ( 1st ` y ) e. _om ) ) |
50 |
43 49
|
mpbird |
|- ( ( X = <. x , y >. /\ ( x e. { (/) } /\ y e. ( _om X. _om ) ) ) -> ( 1st ` ( 2nd ` X ) ) e. _om ) |
51 |
50
|
exlimivv |
|- ( E. x E. y ( X = <. x , y >. /\ ( x e. { (/) } /\ y e. ( _om X. _om ) ) ) -> ( 1st ` ( 2nd ` X ) ) e. _om ) |
52 |
41 51
|
sylbi |
|- ( X e. ( Fmla ` (/) ) -> ( 1st ` ( 2nd ` X ) ) e. _om ) |
53 |
52
|
ad2antll |
|- ( ( a : _om --> M /\ ( M e. V /\ X e. ( Fmla ` (/) ) ) ) -> ( 1st ` ( 2nd ` X ) ) e. _om ) |
54 |
37 53
|
ffvelrnd |
|- ( ( a : _om --> M /\ ( M e. V /\ X e. ( Fmla ` (/) ) ) ) -> ( a ` ( 1st ` ( 2nd ` X ) ) ) e. M ) |
55 |
|
xp2nd |
|- ( y e. ( _om X. _om ) -> ( 2nd ` y ) e. _om ) |
56 |
55
|
ad2antll |
|- ( ( X = <. x , y >. /\ ( x e. { (/) } /\ y e. ( _om X. _om ) ) ) -> ( 2nd ` y ) e. _om ) |
57 |
46
|
fveq2d |
|- ( X = <. x , y >. -> ( 2nd ` ( 2nd ` X ) ) = ( 2nd ` y ) ) |
58 |
57
|
eleq1d |
|- ( X = <. x , y >. -> ( ( 2nd ` ( 2nd ` X ) ) e. _om <-> ( 2nd ` y ) e. _om ) ) |
59 |
58
|
adantr |
|- ( ( X = <. x , y >. /\ ( x e. { (/) } /\ y e. ( _om X. _om ) ) ) -> ( ( 2nd ` ( 2nd ` X ) ) e. _om <-> ( 2nd ` y ) e. _om ) ) |
60 |
56 59
|
mpbird |
|- ( ( X = <. x , y >. /\ ( x e. { (/) } /\ y e. ( _om X. _om ) ) ) -> ( 2nd ` ( 2nd ` X ) ) e. _om ) |
61 |
60
|
exlimivv |
|- ( E. x E. y ( X = <. x , y >. /\ ( x e. { (/) } /\ y e. ( _om X. _om ) ) ) -> ( 2nd ` ( 2nd ` X ) ) e. _om ) |
62 |
41 61
|
sylbi |
|- ( X e. ( Fmla ` (/) ) -> ( 2nd ` ( 2nd ` X ) ) e. _om ) |
63 |
62
|
ad2antll |
|- ( ( a : _om --> M /\ ( M e. V /\ X e. ( Fmla ` (/) ) ) ) -> ( 2nd ` ( 2nd ` X ) ) e. _om ) |
64 |
37 63
|
ffvelrnd |
|- ( ( a : _om --> M /\ ( M e. V /\ X e. ( Fmla ` (/) ) ) ) -> ( a ` ( 2nd ` ( 2nd ` X ) ) ) e. M ) |
65 |
54 64
|
jca |
|- ( ( a : _om --> M /\ ( M e. V /\ X e. ( Fmla ` (/) ) ) ) -> ( ( a ` ( 1st ` ( 2nd ` X ) ) ) e. M /\ ( a ` ( 2nd ` ( 2nd ` X ) ) ) e. M ) ) |
66 |
65
|
ex |
|- ( a : _om --> M -> ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( ( a ` ( 1st ` ( 2nd ` X ) ) ) e. M /\ ( a ` ( 2nd ` ( 2nd ` X ) ) ) e. M ) ) ) |
67 |
36 66
|
syl |
|- ( a e. ( M ^m _om ) -> ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( ( a ` ( 1st ` ( 2nd ` X ) ) ) e. M /\ ( a ` ( 2nd ` ( 2nd ` X ) ) ) e. M ) ) ) |
68 |
67
|
impcom |
|- ( ( ( M e. V /\ X e. ( Fmla ` (/) ) ) /\ a e. ( M ^m _om ) ) -> ( ( a ` ( 1st ` ( 2nd ` X ) ) ) e. M /\ ( a ` ( 2nd ` ( 2nd ` X ) ) ) e. M ) ) |
69 |
|
brinxp |
|- ( ( ( a ` ( 1st ` ( 2nd ` X ) ) ) e. M /\ ( a ` ( 2nd ` ( 2nd ` X ) ) ) e. M ) -> ( ( a ` ( 1st ` ( 2nd ` X ) ) ) _E ( a ` ( 2nd ` ( 2nd ` X ) ) ) <-> ( a ` ( 1st ` ( 2nd ` X ) ) ) ( _E i^i ( M X. M ) ) ( a ` ( 2nd ` ( 2nd ` X ) ) ) ) ) |
70 |
69
|
bicomd |
|- ( ( ( a ` ( 1st ` ( 2nd ` X ) ) ) e. M /\ ( a ` ( 2nd ` ( 2nd ` X ) ) ) e. M ) -> ( ( a ` ( 1st ` ( 2nd ` X ) ) ) ( _E i^i ( M X. M ) ) ( a ` ( 2nd ` ( 2nd ` X ) ) ) <-> ( a ` ( 1st ` ( 2nd ` X ) ) ) _E ( a ` ( 2nd ` ( 2nd ` X ) ) ) ) ) |
71 |
68 70
|
syl |
|- ( ( ( M e. V /\ X e. ( Fmla ` (/) ) ) /\ a e. ( M ^m _om ) ) -> ( ( a ` ( 1st ` ( 2nd ` X ) ) ) ( _E i^i ( M X. M ) ) ( a ` ( 2nd ` ( 2nd ` X ) ) ) <-> ( a ` ( 1st ` ( 2nd ` X ) ) ) _E ( a ` ( 2nd ` ( 2nd ` X ) ) ) ) ) |
72 |
|
fvex |
|- ( a ` ( 2nd ` ( 2nd ` X ) ) ) e. _V |
73 |
72
|
epeli |
|- ( ( a ` ( 1st ` ( 2nd ` X ) ) ) _E ( a ` ( 2nd ` ( 2nd ` X ) ) ) <-> ( a ` ( 1st ` ( 2nd ` X ) ) ) e. ( a ` ( 2nd ` ( 2nd ` X ) ) ) ) |
74 |
71 73
|
bitrdi |
|- ( ( ( M e. V /\ X e. ( Fmla ` (/) ) ) /\ a e. ( M ^m _om ) ) -> ( ( a ` ( 1st ` ( 2nd ` X ) ) ) ( _E i^i ( M X. M ) ) ( a ` ( 2nd ` ( 2nd ` X ) ) ) <-> ( a ` ( 1st ` ( 2nd ` X ) ) ) e. ( a ` ( 2nd ` ( 2nd ` X ) ) ) ) ) |
75 |
74
|
rabbidva |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` X ) ) ) ( _E i^i ( M X. M ) ) ( a ` ( 2nd ` ( 2nd ` X ) ) ) } = { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` X ) ) ) e. ( a ` ( 2nd ` ( 2nd ` X ) ) ) } ) |
76 |
35 75
|
eqtrd |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` (/) ) ` X ) = { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` X ) ) ) e. ( a ` ( 2nd ` ( 2nd ` X ) ) ) } ) |
77 |
29 76
|
eqtrd |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` X ) = { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` X ) ) ) e. ( a ` ( 2nd ` ( 2nd ` X ) ) ) } ) |
78 |
1 77
|
eqtrd |
|- ( ( M e. V /\ X e. ( Fmla ` (/) ) ) -> ( M SatE X ) = { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` X ) ) ) e. ( a ` ( 2nd ` ( 2nd ` X ) ) ) } ) |