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 |
|
oemapval.t |
|- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) } |
5 |
|
oemapval.f |
|- ( ph -> F e. S ) |
6 |
|
oemapval.g |
|- ( ph -> G e. S ) |
7 |
|
oemapvali.r |
|- ( ph -> F T G ) |
8 |
|
oemapvali.x |
|- X = U. { c e. B | ( F ` c ) e. ( G ` c ) } |
9 |
1 2 3 4 5 6 7 8
|
oemapvali |
|- ( ph -> ( X e. B /\ ( F ` X ) e. ( G ` X ) /\ A. w e. B ( X e. w -> ( F ` w ) = ( G ` w ) ) ) ) |
10 |
9
|
simp1d |
|- ( ph -> X e. B ) |
11 |
9
|
simp2d |
|- ( ph -> ( F ` X ) e. ( G ` X ) ) |
12 |
11
|
ne0d |
|- ( ph -> ( G ` X ) =/= (/) ) |
13 |
1 2 3
|
cantnfs |
|- ( ph -> ( G e. S <-> ( G : B --> A /\ G finSupp (/) ) ) ) |
14 |
6 13
|
mpbid |
|- ( ph -> ( G : B --> A /\ G finSupp (/) ) ) |
15 |
14
|
simpld |
|- ( ph -> G : B --> A ) |
16 |
15
|
ffnd |
|- ( ph -> G Fn B ) |
17 |
|
0ex |
|- (/) e. _V |
18 |
17
|
a1i |
|- ( ph -> (/) e. _V ) |
19 |
|
elsuppfn |
|- ( ( G Fn B /\ B e. On /\ (/) e. _V ) -> ( X e. ( G supp (/) ) <-> ( X e. B /\ ( G ` X ) =/= (/) ) ) ) |
20 |
16 3 18 19
|
syl3anc |
|- ( ph -> ( X e. ( G supp (/) ) <-> ( X e. B /\ ( G ` X ) =/= (/) ) ) ) |
21 |
10 12 20
|
mpbir2and |
|- ( ph -> X e. ( G supp (/) ) ) |