Step |
Hyp |
Ref |
Expression |
1 |
|
idinxpresid |
|- ( _I i^i ( A X. A ) ) = ( _I |` A ) |
2 |
1
|
sseq1i |
|- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> ( _I |` A ) C_ ( R i^i ( A X. A ) ) ) |
3 |
|
idrefALT |
|- ( ( _I |` A ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A x ( R i^i ( A X. A ) ) x ) |
4 |
|
brinxp2 |
|- ( x ( R i^i ( A X. A ) ) x <-> ( ( x e. A /\ x e. A ) /\ x R x ) ) |
5 |
|
pm4.24 |
|- ( x e. A <-> ( x e. A /\ x e. A ) ) |
6 |
5
|
anbi1i |
|- ( ( x e. A /\ x R x ) <-> ( ( x e. A /\ x e. A ) /\ x R x ) ) |
7 |
4 6
|
bitr4i |
|- ( x ( R i^i ( A X. A ) ) x <-> ( x e. A /\ x R x ) ) |
8 |
7
|
ralbii |
|- ( A. x e. A x ( R i^i ( A X. A ) ) x <-> A. x e. A ( x e. A /\ x R x ) ) |
9 |
2 3 8
|
3bitri |
|- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A ( x e. A /\ x R x ) ) |
10 |
|
ralanid |
|- ( A. x e. A ( x e. A /\ x R x ) <-> A. x e. A x R x ) |
11 |
9 10
|
bitri |
|- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A x R x ) |