| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cantnfs.s |
|- S = dom ( A CNF B ) |
| 2 |
|
cantnfs.a |
|- ( ph -> A e. On ) |
| 3 |
|
cantnfs.b |
|- ( ph -> B e. On ) |
| 4 |
|
cantnfcl.g |
|- G = OrdIso ( _E , ( F supp (/) ) ) |
| 5 |
|
cantnfcl.f |
|- ( ph -> F e. S ) |
| 6 |
|
suppssdm |
|- ( F supp (/) ) C_ dom F |
| 7 |
1 2 3
|
cantnfs |
|- ( ph -> ( F e. S <-> ( F : B --> A /\ F finSupp (/) ) ) ) |
| 8 |
5 7
|
mpbid |
|- ( ph -> ( F : B --> A /\ F finSupp (/) ) ) |
| 9 |
8
|
simpld |
|- ( ph -> F : B --> A ) |
| 10 |
6 9
|
fssdm |
|- ( ph -> ( F supp (/) ) C_ B ) |
| 11 |
|
onss |
|- ( B e. On -> B C_ On ) |
| 12 |
3 11
|
syl |
|- ( ph -> B C_ On ) |
| 13 |
10 12
|
sstrd |
|- ( ph -> ( F supp (/) ) C_ On ) |
| 14 |
|
epweon |
|- _E We On |
| 15 |
|
wess |
|- ( ( F supp (/) ) C_ On -> ( _E We On -> _E We ( F supp (/) ) ) ) |
| 16 |
13 14 15
|
mpisyl |
|- ( ph -> _E We ( F supp (/) ) ) |
| 17 |
|
ovexd |
|- ( ph -> ( F supp (/) ) e. _V ) |
| 18 |
4
|
oion |
|- ( ( F supp (/) ) e. _V -> dom G e. On ) |
| 19 |
17 18
|
syl |
|- ( ph -> dom G e. On ) |
| 20 |
8
|
simprd |
|- ( ph -> F finSupp (/) ) |
| 21 |
20
|
fsuppimpd |
|- ( ph -> ( F supp (/) ) e. Fin ) |
| 22 |
4
|
oien |
|- ( ( ( F supp (/) ) e. _V /\ _E We ( F supp (/) ) ) -> dom G ~~ ( F supp (/) ) ) |
| 23 |
17 16 22
|
syl2anc |
|- ( ph -> dom G ~~ ( F supp (/) ) ) |
| 24 |
|
enfii |
|- ( ( ( F supp (/) ) e. Fin /\ dom G ~~ ( F supp (/) ) ) -> dom G e. Fin ) |
| 25 |
21 23 24
|
syl2anc |
|- ( ph -> dom G e. Fin ) |
| 26 |
19 25
|
elind |
|- ( ph -> dom G e. ( On i^i Fin ) ) |
| 27 |
|
onfin2 |
|- _om = ( On i^i Fin ) |
| 28 |
26 27
|
eleqtrrdi |
|- ( ph -> dom G e. _om ) |
| 29 |
16 28
|
jca |
|- ( ph -> ( _E We ( F supp (/) ) /\ dom G e. _om ) ) |