Step |
Hyp |
Ref |
Expression |
1 |
|
df-pprod |
|- pprod ( A , B ) = ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) |
2 |
|
txprel |
|- Rel ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) |
3 |
|
txpss3v |
|- ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) C_ ( _V X. ( _V X. _V ) ) |
4 |
3
|
sseli |
|- ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> <. x , y >. e. ( _V X. ( _V X. _V ) ) ) |
5 |
|
opelxp2 |
|- ( <. x , y >. e. ( _V X. ( _V X. _V ) ) -> y e. ( _V X. _V ) ) |
6 |
4 5
|
syl |
|- ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> y e. ( _V X. _V ) ) |
7 |
|
elvv |
|- ( y e. ( _V X. _V ) <-> E. z E. w y = <. z , w >. ) |
8 |
|
opeq2 |
|- ( y = <. z , w >. -> <. x , y >. = <. x , <. z , w >. >. ) |
9 |
8
|
eleq1d |
|- ( y = <. z , w >. -> ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) <-> <. x , <. z , w >. >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) ) ) |
10 |
|
df-br |
|- ( x ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) <. z , w >. <-> <. x , <. z , w >. >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) ) |
11 |
|
vex |
|- x e. _V |
12 |
|
vex |
|- z e. _V |
13 |
|
vex |
|- w e. _V |
14 |
11 12 13
|
brtxp |
|- ( x ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) <. z , w >. <-> ( x ( A o. ( 1st |` ( _V X. _V ) ) ) z /\ x ( B o. ( 2nd |` ( _V X. _V ) ) ) w ) ) |
15 |
11 12
|
brco |
|- ( x ( A o. ( 1st |` ( _V X. _V ) ) ) z <-> E. y ( x ( 1st |` ( _V X. _V ) ) y /\ y A z ) ) |
16 |
|
vex |
|- y e. _V |
17 |
16
|
brresi |
|- ( x ( 1st |` ( _V X. _V ) ) y <-> ( x e. ( _V X. _V ) /\ x 1st y ) ) |
18 |
17
|
simplbi |
|- ( x ( 1st |` ( _V X. _V ) ) y -> x e. ( _V X. _V ) ) |
19 |
18
|
adantr |
|- ( ( x ( 1st |` ( _V X. _V ) ) y /\ y A z ) -> x e. ( _V X. _V ) ) |
20 |
19
|
exlimiv |
|- ( E. y ( x ( 1st |` ( _V X. _V ) ) y /\ y A z ) -> x e. ( _V X. _V ) ) |
21 |
15 20
|
sylbi |
|- ( x ( A o. ( 1st |` ( _V X. _V ) ) ) z -> x e. ( _V X. _V ) ) |
22 |
21
|
adantr |
|- ( ( x ( A o. ( 1st |` ( _V X. _V ) ) ) z /\ x ( B o. ( 2nd |` ( _V X. _V ) ) ) w ) -> x e. ( _V X. _V ) ) |
23 |
14 22
|
sylbi |
|- ( x ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) <. z , w >. -> x e. ( _V X. _V ) ) |
24 |
10 23
|
sylbir |
|- ( <. x , <. z , w >. >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> x e. ( _V X. _V ) ) |
25 |
9 24
|
syl6bi |
|- ( y = <. z , w >. -> ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> x e. ( _V X. _V ) ) ) |
26 |
25
|
exlimivv |
|- ( E. z E. w y = <. z , w >. -> ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> x e. ( _V X. _V ) ) ) |
27 |
7 26
|
sylbi |
|- ( y e. ( _V X. _V ) -> ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> x e. ( _V X. _V ) ) ) |
28 |
6 27
|
mpcom |
|- ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> x e. ( _V X. _V ) ) |
29 |
28 6
|
opelxpd |
|- ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> <. x , y >. e. ( ( _V X. _V ) X. ( _V X. _V ) ) ) |
30 |
2 29
|
relssi |
|- ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) C_ ( ( _V X. _V ) X. ( _V X. _V ) ) |
31 |
1 30
|
eqsstri |
|- pprod ( A , B ) C_ ( ( _V X. _V ) X. ( _V X. _V ) ) |