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 |
|
df-inf |
|- inf ( B , A , R ) = sup ( B , A , `' R ) |
4 |
3
|
breq1i |
|- ( inf ( B , A , R ) R C <-> sup ( B , A , `' R ) R C ) |
5 |
|
simpr |
|- ( ( ph /\ C e. A ) -> C e. A ) |
6 |
|
cnvso |
|- ( R Or A <-> `' R Or A ) |
7 |
1 6
|
sylib |
|- ( ph -> `' R Or A ) |
8 |
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 ) ) ) |
9 |
7 8
|
supcl |
|- ( ph -> sup ( B , A , `' R ) e. A ) |
10 |
9
|
adantr |
|- ( ( ph /\ C e. A ) -> sup ( B , A , `' R ) e. A ) |
11 |
|
brcnvg |
|- ( ( C e. A /\ sup ( B , A , `' R ) e. A ) -> ( C `' R sup ( B , A , `' R ) <-> sup ( B , A , `' R ) R C ) ) |
12 |
11
|
bicomd |
|- ( ( C e. A /\ sup ( B , A , `' R ) e. A ) -> ( sup ( B , A , `' R ) R C <-> C `' R sup ( B , A , `' R ) ) ) |
13 |
5 10 12
|
syl2anc |
|- ( ( ph /\ C e. A ) -> ( sup ( B , A , `' R ) R C <-> C `' R sup ( B , A , `' R ) ) ) |
14 |
4 13
|
syl5bb |
|- ( ( ph /\ C e. A ) -> ( inf ( B , A , R ) R C <-> C `' R sup ( B , A , `' R ) ) ) |
15 |
7 8
|
suplub |
|- ( ph -> ( ( C e. A /\ C `' R sup ( B , A , `' R ) ) -> E. z e. B C `' R z ) ) |
16 |
15
|
expdimp |
|- ( ( ph /\ C e. A ) -> ( C `' R sup ( B , A , `' R ) -> E. z e. B C `' R z ) ) |
17 |
|
vex |
|- z e. _V |
18 |
|
brcnvg |
|- ( ( C e. A /\ z e. _V ) -> ( C `' R z <-> z R C ) ) |
19 |
5 17 18
|
sylancl |
|- ( ( ph /\ C e. A ) -> ( C `' R z <-> z R C ) ) |
20 |
19
|
rexbidv |
|- ( ( ph /\ C e. A ) -> ( E. z e. B C `' R z <-> E. z e. B z R C ) ) |
21 |
16 20
|
sylibd |
|- ( ( ph /\ C e. A ) -> ( C `' R sup ( B , A , `' R ) -> E. z e. B z R C ) ) |
22 |
14 21
|
sylbid |
|- ( ( ph /\ C e. A ) -> ( inf ( B , A , R ) R C -> E. z e. B z R C ) ) |
23 |
22
|
expimpd |
|- ( ph -> ( ( C e. A /\ inf ( B , A , R ) R C ) -> E. z e. B z R C ) ) |