Step |
Hyp |
Ref |
Expression |
1 |
|
eloprabi.1 |
|- ( x = ( 1st ` ( 1st ` A ) ) -> ( ph <-> ps ) ) |
2 |
|
eloprabi.2 |
|- ( y = ( 2nd ` ( 1st ` A ) ) -> ( ps <-> ch ) ) |
3 |
|
eloprabi.3 |
|- ( z = ( 2nd ` A ) -> ( ch <-> th ) ) |
4 |
|
eqeq1 |
|- ( w = A -> ( w = <. <. x , y >. , z >. <-> A = <. <. x , y >. , z >. ) ) |
5 |
4
|
anbi1d |
|- ( w = A -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( A = <. <. x , y >. , z >. /\ ph ) ) ) |
6 |
5
|
3exbidv |
|- ( w = A -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. x E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) ) ) |
7 |
|
df-oprab |
|- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } |
8 |
6 7
|
elab2g |
|- ( A e. { <. <. x , y >. , z >. | ph } -> ( A e. { <. <. x , y >. , z >. | ph } <-> E. x E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) ) ) |
9 |
8
|
ibi |
|- ( A e. { <. <. x , y >. , z >. | ph } -> E. x E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) ) |
10 |
|
opex |
|- <. x , y >. e. _V |
11 |
|
vex |
|- z e. _V |
12 |
10 11
|
op1std |
|- ( A = <. <. x , y >. , z >. -> ( 1st ` A ) = <. x , y >. ) |
13 |
12
|
fveq2d |
|- ( A = <. <. x , y >. , z >. -> ( 1st ` ( 1st ` A ) ) = ( 1st ` <. x , y >. ) ) |
14 |
|
vex |
|- x e. _V |
15 |
|
vex |
|- y e. _V |
16 |
14 15
|
op1st |
|- ( 1st ` <. x , y >. ) = x |
17 |
13 16
|
eqtr2di |
|- ( A = <. <. x , y >. , z >. -> x = ( 1st ` ( 1st ` A ) ) ) |
18 |
17 1
|
syl |
|- ( A = <. <. x , y >. , z >. -> ( ph <-> ps ) ) |
19 |
12
|
fveq2d |
|- ( A = <. <. x , y >. , z >. -> ( 2nd ` ( 1st ` A ) ) = ( 2nd ` <. x , y >. ) ) |
20 |
14 15
|
op2nd |
|- ( 2nd ` <. x , y >. ) = y |
21 |
19 20
|
eqtr2di |
|- ( A = <. <. x , y >. , z >. -> y = ( 2nd ` ( 1st ` A ) ) ) |
22 |
21 2
|
syl |
|- ( A = <. <. x , y >. , z >. -> ( ps <-> ch ) ) |
23 |
10 11
|
op2ndd |
|- ( A = <. <. x , y >. , z >. -> ( 2nd ` A ) = z ) |
24 |
23
|
eqcomd |
|- ( A = <. <. x , y >. , z >. -> z = ( 2nd ` A ) ) |
25 |
24 3
|
syl |
|- ( A = <. <. x , y >. , z >. -> ( ch <-> th ) ) |
26 |
18 22 25
|
3bitrd |
|- ( A = <. <. x , y >. , z >. -> ( ph <-> th ) ) |
27 |
26
|
biimpa |
|- ( ( A = <. <. x , y >. , z >. /\ ph ) -> th ) |
28 |
27
|
exlimiv |
|- ( E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th ) |
29 |
28
|
exlimiv |
|- ( E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th ) |
30 |
29
|
exlimiv |
|- ( E. x E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th ) |
31 |
9 30
|
syl |
|- ( A e. { <. <. x , y >. , z >. | ph } -> th ) |