Step |
Hyp |
Ref |
Expression |
1 |
|
sategoelfvb.s |
|- E = ( M SatE ( A e.g B ) ) |
2 |
|
ovexd |
|- ( ( A e. _om /\ B e. _om ) -> ( A e.g B ) e. _V ) |
3 |
|
simpl |
|- ( ( A e. _om /\ B e. _om ) -> A e. _om ) |
4 |
|
opeq1 |
|- ( a = A -> <. a , b >. = <. A , b >. ) |
5 |
4
|
opeq2d |
|- ( a = A -> <. (/) , <. a , b >. >. = <. (/) , <. A , b >. >. ) |
6 |
5
|
eqeq2d |
|- ( a = A -> ( <. (/) , <. A , B >. >. = <. (/) , <. a , b >. >. <-> <. (/) , <. A , B >. >. = <. (/) , <. A , b >. >. ) ) |
7 |
6
|
rexbidv |
|- ( a = A -> ( E. b e. _om <. (/) , <. A , B >. >. = <. (/) , <. a , b >. >. <-> E. b e. _om <. (/) , <. A , B >. >. = <. (/) , <. A , b >. >. ) ) |
8 |
7
|
adantl |
|- ( ( ( A e. _om /\ B e. _om ) /\ a = A ) -> ( E. b e. _om <. (/) , <. A , B >. >. = <. (/) , <. a , b >. >. <-> E. b e. _om <. (/) , <. A , B >. >. = <. (/) , <. A , b >. >. ) ) |
9 |
|
simpr |
|- ( ( A e. _om /\ B e. _om ) -> B e. _om ) |
10 |
|
opeq2 |
|- ( b = B -> <. A , b >. = <. A , B >. ) |
11 |
10
|
opeq2d |
|- ( b = B -> <. (/) , <. A , b >. >. = <. (/) , <. A , B >. >. ) |
12 |
11
|
eqeq2d |
|- ( b = B -> ( <. (/) , <. A , B >. >. = <. (/) , <. A , b >. >. <-> <. (/) , <. A , B >. >. = <. (/) , <. A , B >. >. ) ) |
13 |
12
|
adantl |
|- ( ( ( A e. _om /\ B e. _om ) /\ b = B ) -> ( <. (/) , <. A , B >. >. = <. (/) , <. A , b >. >. <-> <. (/) , <. A , B >. >. = <. (/) , <. A , B >. >. ) ) |
14 |
|
eqidd |
|- ( ( A e. _om /\ B e. _om ) -> <. (/) , <. A , B >. >. = <. (/) , <. A , B >. >. ) |
15 |
9 13 14
|
rspcedvd |
|- ( ( A e. _om /\ B e. _om ) -> E. b e. _om <. (/) , <. A , B >. >. = <. (/) , <. A , b >. >. ) |
16 |
3 8 15
|
rspcedvd |
|- ( ( A e. _om /\ B e. _om ) -> E. a e. _om E. b e. _om <. (/) , <. A , B >. >. = <. (/) , <. a , b >. >. ) |
17 |
|
goel |
|- ( ( A e. _om /\ B e. _om ) -> ( A e.g B ) = <. (/) , <. A , B >. >. ) |
18 |
|
goel |
|- ( ( a e. _om /\ b e. _om ) -> ( a e.g b ) = <. (/) , <. a , b >. >. ) |
19 |
17 18
|
eqeqan12d |
|- ( ( ( A e. _om /\ B e. _om ) /\ ( a e. _om /\ b e. _om ) ) -> ( ( A e.g B ) = ( a e.g b ) <-> <. (/) , <. A , B >. >. = <. (/) , <. a , b >. >. ) ) |
20 |
19
|
2rexbidva |
|- ( ( A e. _om /\ B e. _om ) -> ( E. a e. _om E. b e. _om ( A e.g B ) = ( a e.g b ) <-> E. a e. _om E. b e. _om <. (/) , <. A , B >. >. = <. (/) , <. a , b >. >. ) ) |
21 |
16 20
|
mpbird |
|- ( ( A e. _om /\ B e. _om ) -> E. a e. _om E. b e. _om ( A e.g B ) = ( a e.g b ) ) |
22 |
|
eqeq1 |
|- ( x = ( A e.g B ) -> ( x = ( a e.g b ) <-> ( A e.g B ) = ( a e.g b ) ) ) |
23 |
22
|
2rexbidv |
|- ( x = ( A e.g B ) -> ( E. a e. _om E. b e. _om x = ( a e.g b ) <-> E. a e. _om E. b e. _om ( A e.g B ) = ( a e.g b ) ) ) |
24 |
|
fmla0 |
|- ( Fmla ` (/) ) = { x e. _V | E. a e. _om E. b e. _om x = ( a e.g b ) } |
25 |
23 24
|
elrab2 |
|- ( ( A e.g B ) e. ( Fmla ` (/) ) <-> ( ( A e.g B ) e. _V /\ E. a e. _om E. b e. _om ( A e.g B ) = ( a e.g b ) ) ) |
26 |
2 21 25
|
sylanbrc |
|- ( ( A e. _om /\ B e. _om ) -> ( A e.g B ) e. ( Fmla ` (/) ) ) |
27 |
|
satefvfmla0 |
|- ( ( M e. V /\ ( A e.g B ) e. ( Fmla ` (/) ) ) -> ( M SatE ( A e.g B ) ) = { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( a ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) } ) |
28 |
26 27
|
sylan2 |
|- ( ( M e. V /\ ( A e. _om /\ B e. _om ) ) -> ( M SatE ( A e.g B ) ) = { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( a ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) } ) |
29 |
1 28
|
syl5eq |
|- ( ( M e. V /\ ( A e. _om /\ B e. _om ) ) -> E = { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( a ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) } ) |
30 |
29
|
eleq2d |
|- ( ( M e. V /\ ( A e. _om /\ B e. _om ) ) -> ( S e. E <-> S e. { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( a ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) } ) ) |
31 |
|
fveq1 |
|- ( a = S -> ( a ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) = ( S ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) ) |
32 |
|
fveq1 |
|- ( a = S -> ( a ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) = ( S ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) ) |
33 |
31 32
|
eleq12d |
|- ( a = S -> ( ( a ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( a ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) <-> ( S ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( S ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) ) ) |
34 |
33
|
elrab |
|- ( S e. { a e. ( M ^m _om ) | ( a ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( a ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) } <-> ( S e. ( M ^m _om ) /\ ( S ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( S ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) ) ) |
35 |
30 34
|
bitrdi |
|- ( ( M e. V /\ ( A e. _om /\ B e. _om ) ) -> ( S e. E <-> ( S e. ( M ^m _om ) /\ ( S ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( S ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) ) ) ) |
36 |
17
|
fveq2d |
|- ( ( A e. _om /\ B e. _om ) -> ( 2nd ` ( A e.g B ) ) = ( 2nd ` <. (/) , <. A , B >. >. ) ) |
37 |
36
|
fveq2d |
|- ( ( A e. _om /\ B e. _om ) -> ( 1st ` ( 2nd ` ( A e.g B ) ) ) = ( 1st ` ( 2nd ` <. (/) , <. A , B >. >. ) ) ) |
38 |
|
0ex |
|- (/) e. _V |
39 |
|
opex |
|- <. A , B >. e. _V |
40 |
38 39
|
op2nd |
|- ( 2nd ` <. (/) , <. A , B >. >. ) = <. A , B >. |
41 |
40
|
fveq2i |
|- ( 1st ` ( 2nd ` <. (/) , <. A , B >. >. ) ) = ( 1st ` <. A , B >. ) |
42 |
|
op1stg |
|- ( ( A e. _om /\ B e. _om ) -> ( 1st ` <. A , B >. ) = A ) |
43 |
41 42
|
syl5eq |
|- ( ( A e. _om /\ B e. _om ) -> ( 1st ` ( 2nd ` <. (/) , <. A , B >. >. ) ) = A ) |
44 |
37 43
|
eqtrd |
|- ( ( A e. _om /\ B e. _om ) -> ( 1st ` ( 2nd ` ( A e.g B ) ) ) = A ) |
45 |
44
|
fveq2d |
|- ( ( A e. _om /\ B e. _om ) -> ( S ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) = ( S ` A ) ) |
46 |
36
|
fveq2d |
|- ( ( A e. _om /\ B e. _om ) -> ( 2nd ` ( 2nd ` ( A e.g B ) ) ) = ( 2nd ` ( 2nd ` <. (/) , <. A , B >. >. ) ) ) |
47 |
40
|
fveq2i |
|- ( 2nd ` ( 2nd ` <. (/) , <. A , B >. >. ) ) = ( 2nd ` <. A , B >. ) |
48 |
|
op2ndg |
|- ( ( A e. _om /\ B e. _om ) -> ( 2nd ` <. A , B >. ) = B ) |
49 |
47 48
|
syl5eq |
|- ( ( A e. _om /\ B e. _om ) -> ( 2nd ` ( 2nd ` <. (/) , <. A , B >. >. ) ) = B ) |
50 |
46 49
|
eqtrd |
|- ( ( A e. _om /\ B e. _om ) -> ( 2nd ` ( 2nd ` ( A e.g B ) ) ) = B ) |
51 |
50
|
fveq2d |
|- ( ( A e. _om /\ B e. _om ) -> ( S ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) = ( S ` B ) ) |
52 |
45 51
|
eleq12d |
|- ( ( A e. _om /\ B e. _om ) -> ( ( S ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( S ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) <-> ( S ` A ) e. ( S ` B ) ) ) |
53 |
52
|
adantl |
|- ( ( M e. V /\ ( A e. _om /\ B e. _om ) ) -> ( ( S ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( S ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) <-> ( S ` A ) e. ( S ` B ) ) ) |
54 |
53
|
anbi2d |
|- ( ( M e. V /\ ( A e. _om /\ B e. _om ) ) -> ( ( S e. ( M ^m _om ) /\ ( S ` ( 1st ` ( 2nd ` ( A e.g B ) ) ) ) e. ( S ` ( 2nd ` ( 2nd ` ( A e.g B ) ) ) ) ) <-> ( S e. ( M ^m _om ) /\ ( S ` A ) e. ( S ` B ) ) ) ) |
55 |
35 54
|
bitrd |
|- ( ( M e. V /\ ( A e. _om /\ B e. _om ) ) -> ( S e. E <-> ( S e. ( M ^m _om ) /\ ( S ` A ) e. ( S ` B ) ) ) ) |