| Step |
Hyp |
Ref |
Expression |
| 1 |
|
infcl.1 |
|- ( ph -> R Or A ) |
| 2 |
|
infcl.2 |
|- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) ) |
| 3 |
|
infglbb.3 |
|- ( ph -> B C_ A ) |
| 4 |
|
df-inf |
|- inf ( B , A , R ) = sup ( B , A , `' R ) |
| 5 |
4
|
breq1i |
|- ( inf ( B , A , R ) R C <-> sup ( B , A , `' R ) R C ) |
| 6 |
|
simpr |
|- ( ( ph /\ C e. A ) -> C e. A ) |
| 7 |
|
cnvso |
|- ( R Or A <-> `' R Or A ) |
| 8 |
1 7
|
sylib |
|- ( ph -> `' R Or A ) |
| 9 |
1 2
|
infcllem |
|- ( ph -> E. x e. A ( A. y e. B -. x `' R y /\ A. y e. A ( y `' R x -> E. z e. B y `' R z ) ) ) |
| 10 |
8 9
|
supcl |
|- ( ph -> sup ( B , A , `' R ) e. A ) |
| 11 |
10
|
adantr |
|- ( ( ph /\ C e. A ) -> sup ( B , A , `' R ) e. A ) |
| 12 |
|
brcnvg |
|- ( ( C e. A /\ sup ( B , A , `' R ) e. A ) -> ( C `' R sup ( B , A , `' R ) <-> sup ( B , A , `' R ) R C ) ) |
| 13 |
12
|
bicomd |
|- ( ( C e. A /\ sup ( B , A , `' R ) e. A ) -> ( sup ( B , A , `' R ) R C <-> C `' R sup ( B , A , `' R ) ) ) |
| 14 |
6 11 13
|
syl2anc |
|- ( ( ph /\ C e. A ) -> ( sup ( B , A , `' R ) R C <-> C `' R sup ( B , A , `' R ) ) ) |
| 15 |
8 9 3
|
suplub2 |
|- ( ( ph /\ C e. A ) -> ( C `' R sup ( B , A , `' R ) <-> E. z e. B C `' R z ) ) |
| 16 |
|
vex |
|- z e. _V |
| 17 |
|
brcnvg |
|- ( ( C e. A /\ z e. _V ) -> ( C `' R z <-> z R C ) ) |
| 18 |
6 16 17
|
sylancl |
|- ( ( ph /\ C e. A ) -> ( C `' R z <-> z R C ) ) |
| 19 |
18
|
rexbidv |
|- ( ( ph /\ C e. A ) -> ( E. z e. B C `' R z <-> E. z e. B z R C ) ) |
| 20 |
14 15 19
|
3bitrd |
|- ( ( ph /\ C e. A ) -> ( sup ( B , A , `' R ) R C <-> E. z e. B z R C ) ) |
| 21 |
5 20
|
bitrid |
|- ( ( ph /\ C e. A ) -> ( inf ( B , A , R ) R C <-> E. z e. B z R C ) ) |