Step |
Hyp |
Ref |
Expression |
1 |
|
brpprod3.1 |
|- X e. _V |
2 |
|
brpprod3.2 |
|- Y e. _V |
3 |
|
brpprod3.3 |
|- Z e. _V |
4 |
|
pprodss4v |
|- pprod ( R , S ) C_ ( ( _V X. _V ) X. ( _V X. _V ) ) |
5 |
4
|
brel |
|- ( <. X , Y >. pprod ( R , S ) Z -> ( <. X , Y >. e. ( _V X. _V ) /\ Z e. ( _V X. _V ) ) ) |
6 |
5
|
simprd |
|- ( <. X , Y >. pprod ( R , S ) Z -> Z e. ( _V X. _V ) ) |
7 |
|
elvv |
|- ( Z e. ( _V X. _V ) <-> E. z E. w Z = <. z , w >. ) |
8 |
6 7
|
sylib |
|- ( <. X , Y >. pprod ( R , S ) Z -> E. z E. w Z = <. z , w >. ) |
9 |
8
|
pm4.71ri |
|- ( <. X , Y >. pprod ( R , S ) Z <-> ( E. z E. w Z = <. z , w >. /\ <. X , Y >. pprod ( R , S ) Z ) ) |
10 |
|
19.41vv |
|- ( E. z E. w ( Z = <. z , w >. /\ <. X , Y >. pprod ( R , S ) Z ) <-> ( E. z E. w Z = <. z , w >. /\ <. X , Y >. pprod ( R , S ) Z ) ) |
11 |
9 10
|
bitr4i |
|- ( <. X , Y >. pprod ( R , S ) Z <-> E. z E. w ( Z = <. z , w >. /\ <. X , Y >. pprod ( R , S ) Z ) ) |
12 |
|
breq2 |
|- ( Z = <. z , w >. -> ( <. X , Y >. pprod ( R , S ) Z <-> <. X , Y >. pprod ( R , S ) <. z , w >. ) ) |
13 |
12
|
pm5.32i |
|- ( ( Z = <. z , w >. /\ <. X , Y >. pprod ( R , S ) Z ) <-> ( Z = <. z , w >. /\ <. X , Y >. pprod ( R , S ) <. z , w >. ) ) |
14 |
13
|
2exbii |
|- ( E. z E. w ( Z = <. z , w >. /\ <. X , Y >. pprod ( R , S ) Z ) <-> E. z E. w ( Z = <. z , w >. /\ <. X , Y >. pprod ( R , S ) <. z , w >. ) ) |
15 |
|
vex |
|- z e. _V |
16 |
|
vex |
|- w e. _V |
17 |
1 2 15 16
|
brpprod |
|- ( <. X , Y >. pprod ( R , S ) <. z , w >. <-> ( X R z /\ Y S w ) ) |
18 |
17
|
anbi2i |
|- ( ( Z = <. z , w >. /\ <. X , Y >. pprod ( R , S ) <. z , w >. ) <-> ( Z = <. z , w >. /\ ( X R z /\ Y S w ) ) ) |
19 |
|
3anass |
|- ( ( Z = <. z , w >. /\ X R z /\ Y S w ) <-> ( Z = <. z , w >. /\ ( X R z /\ Y S w ) ) ) |
20 |
18 19
|
bitr4i |
|- ( ( Z = <. z , w >. /\ <. X , Y >. pprod ( R , S ) <. z , w >. ) <-> ( Z = <. z , w >. /\ X R z /\ Y S w ) ) |
21 |
20
|
2exbii |
|- ( E. z E. w ( Z = <. z , w >. /\ <. X , Y >. pprod ( R , S ) <. z , w >. ) <-> E. z E. w ( Z = <. z , w >. /\ X R z /\ Y S w ) ) |
22 |
11 14 21
|
3bitri |
|- ( <. X , Y >. pprod ( R , S ) Z <-> E. z E. w ( Z = <. z , w >. /\ X R z /\ Y S w ) ) |