| Step |
Hyp |
Ref |
Expression |
| 1 |
|
joindmss.b |
|- B = ( Base ` K ) |
| 2 |
|
joindmss.j |
|- .\/ = ( join ` K ) |
| 3 |
|
joindmss.k |
|- ( ph -> K e. V ) |
| 4 |
|
relopabv |
|- Rel { <. x , y >. | { x , y } e. dom ( lub ` K ) } |
| 5 |
|
eqid |
|- ( lub ` K ) = ( lub ` K ) |
| 6 |
5 2
|
joindm |
|- ( K e. V -> dom .\/ = { <. x , y >. | { x , y } e. dom ( lub ` K ) } ) |
| 7 |
3 6
|
syl |
|- ( ph -> dom .\/ = { <. x , y >. | { x , y } e. dom ( lub ` K ) } ) |
| 8 |
7
|
releqd |
|- ( ph -> ( Rel dom .\/ <-> Rel { <. x , y >. | { x , y } e. dom ( lub ` K ) } ) ) |
| 9 |
4 8
|
mpbiri |
|- ( ph -> Rel dom .\/ ) |
| 10 |
|
vex |
|- x e. _V |
| 11 |
10
|
a1i |
|- ( ph -> x e. _V ) |
| 12 |
|
vex |
|- y e. _V |
| 13 |
12
|
a1i |
|- ( ph -> y e. _V ) |
| 14 |
5 2 3 11 13
|
joindef |
|- ( ph -> ( <. x , y >. e. dom .\/ <-> { x , y } e. dom ( lub ` K ) ) ) |
| 15 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
| 16 |
3
|
adantr |
|- ( ( ph /\ { x , y } e. dom ( lub ` K ) ) -> K e. V ) |
| 17 |
|
simpr |
|- ( ( ph /\ { x , y } e. dom ( lub ` K ) ) -> { x , y } e. dom ( lub ` K ) ) |
| 18 |
1 15 5 16 17
|
lubelss |
|- ( ( ph /\ { x , y } e. dom ( lub ` K ) ) -> { x , y } C_ B ) |
| 19 |
18
|
ex |
|- ( ph -> ( { x , y } e. dom ( lub ` K ) -> { x , y } C_ B ) ) |
| 20 |
10 12
|
prss |
|- ( ( x e. B /\ y e. B ) <-> { x , y } C_ B ) |
| 21 |
|
opelxpi |
|- ( ( x e. B /\ y e. B ) -> <. x , y >. e. ( B X. B ) ) |
| 22 |
20 21
|
sylbir |
|- ( { x , y } C_ B -> <. x , y >. e. ( B X. B ) ) |
| 23 |
19 22
|
syl6 |
|- ( ph -> ( { x , y } e. dom ( lub ` K ) -> <. x , y >. e. ( B X. B ) ) ) |
| 24 |
14 23
|
sylbid |
|- ( ph -> ( <. x , y >. e. dom .\/ -> <. x , y >. e. ( B X. B ) ) ) |
| 25 |
9 24
|
relssdv |
|- ( ph -> dom .\/ C_ ( B X. B ) ) |