Step |
Hyp |
Ref |
Expression |
1 |
|
dvdsr.1 |
|- B = ( Base ` R ) |
2 |
|
dvdsr.2 |
|- .|| = ( ||r ` R ) |
3 |
|
dvdsr.3 |
|- .x. = ( .r ` R ) |
4 |
2
|
reldvdsr |
|- Rel .|| |
5 |
4
|
brrelex12i |
|- ( X .|| Y -> ( X e. _V /\ Y e. _V ) ) |
6 |
|
elex |
|- ( X e. B -> X e. _V ) |
7 |
|
id |
|- ( ( z .x. X ) = Y -> ( z .x. X ) = Y ) |
8 |
|
ovex |
|- ( z .x. X ) e. _V |
9 |
7 8
|
eqeltrrdi |
|- ( ( z .x. X ) = Y -> Y e. _V ) |
10 |
9
|
rexlimivw |
|- ( E. z e. B ( z .x. X ) = Y -> Y e. _V ) |
11 |
6 10
|
anim12i |
|- ( ( X e. B /\ E. z e. B ( z .x. X ) = Y ) -> ( X e. _V /\ Y e. _V ) ) |
12 |
|
simpl |
|- ( ( x = X /\ y = Y ) -> x = X ) |
13 |
12
|
eleq1d |
|- ( ( x = X /\ y = Y ) -> ( x e. B <-> X e. B ) ) |
14 |
12
|
oveq2d |
|- ( ( x = X /\ y = Y ) -> ( z .x. x ) = ( z .x. X ) ) |
15 |
|
simpr |
|- ( ( x = X /\ y = Y ) -> y = Y ) |
16 |
14 15
|
eqeq12d |
|- ( ( x = X /\ y = Y ) -> ( ( z .x. x ) = y <-> ( z .x. X ) = Y ) ) |
17 |
16
|
rexbidv |
|- ( ( x = X /\ y = Y ) -> ( E. z e. B ( z .x. x ) = y <-> E. z e. B ( z .x. X ) = Y ) ) |
18 |
13 17
|
anbi12d |
|- ( ( x = X /\ y = Y ) -> ( ( x e. B /\ E. z e. B ( z .x. x ) = y ) <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) ) |
19 |
1 2 3
|
dvdsrval |
|- .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } |
20 |
18 19
|
brabga |
|- ( ( X e. _V /\ Y e. _V ) -> ( X .|| Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) ) |
21 |
5 11 20
|
pm5.21nii |
|- ( X .|| Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) |