Step |
Hyp |
Ref |
Expression |
1 |
|
cnfcom.s |
|- S = dom ( _om CNF A ) |
2 |
|
cnfcom.a |
|- ( ph -> A e. On ) |
3 |
|
cnfcom.b |
|- ( ph -> B e. ( _om ^o A ) ) |
4 |
|
cnfcom.f |
|- F = ( `' ( _om CNF A ) ` B ) |
5 |
|
cnfcom.g |
|- G = OrdIso ( _E , ( F supp (/) ) ) |
6 |
|
cnfcom.h |
|- H = seqom ( ( k e. _V , z e. _V |-> ( M +o z ) ) , (/) ) |
7 |
|
cnfcom.t |
|- T = seqom ( ( k e. _V , f e. _V |-> K ) , (/) ) |
8 |
|
cnfcom.m |
|- M = ( ( _om ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) |
9 |
|
cnfcom.k |
|- K = ( ( x e. M |-> ( dom f +o x ) ) u. `' ( x e. dom f |-> ( M +o x ) ) ) |
10 |
|
cnfcom.w |
|- W = ( G ` U. dom G ) |
11 |
|
cnfcom2.1 |
|- ( ph -> (/) e. B ) |
12 |
|
n0i |
|- ( (/) e. B -> -. B = (/) ) |
13 |
11 12
|
syl |
|- ( ph -> -. B = (/) ) |
14 |
|
omelon |
|- _om e. On |
15 |
14
|
a1i |
|- ( ph -> _om e. On ) |
16 |
1 15 2
|
cantnff1o |
|- ( ph -> ( _om CNF A ) : S -1-1-onto-> ( _om ^o A ) ) |
17 |
|
f1ocnv |
|- ( ( _om CNF A ) : S -1-1-onto-> ( _om ^o A ) -> `' ( _om CNF A ) : ( _om ^o A ) -1-1-onto-> S ) |
18 |
|
f1of |
|- ( `' ( _om CNF A ) : ( _om ^o A ) -1-1-onto-> S -> `' ( _om CNF A ) : ( _om ^o A ) --> S ) |
19 |
16 17 18
|
3syl |
|- ( ph -> `' ( _om CNF A ) : ( _om ^o A ) --> S ) |
20 |
19 3
|
ffvelrnd |
|- ( ph -> ( `' ( _om CNF A ) ` B ) e. S ) |
21 |
4 20
|
eqeltrid |
|- ( ph -> F e. S ) |
22 |
1 15 2
|
cantnfs |
|- ( ph -> ( F e. S <-> ( F : A --> _om /\ F finSupp (/) ) ) ) |
23 |
21 22
|
mpbid |
|- ( ph -> ( F : A --> _om /\ F finSupp (/) ) ) |
24 |
23
|
simpld |
|- ( ph -> F : A --> _om ) |
25 |
24
|
adantr |
|- ( ( ph /\ dom G = (/) ) -> F : A --> _om ) |
26 |
25
|
feqmptd |
|- ( ( ph /\ dom G = (/) ) -> F = ( x e. A |-> ( F ` x ) ) ) |
27 |
|
dif0 |
|- ( A \ (/) ) = A |
28 |
27
|
eleq2i |
|- ( x e. ( A \ (/) ) <-> x e. A ) |
29 |
|
simpr |
|- ( ( ph /\ dom G = (/) ) -> dom G = (/) ) |
30 |
|
ovexd |
|- ( ph -> ( F supp (/) ) e. _V ) |
31 |
1 15 2 5 21
|
cantnfcl |
|- ( ph -> ( _E We ( F supp (/) ) /\ dom G e. _om ) ) |
32 |
31
|
simpld |
|- ( ph -> _E We ( F supp (/) ) ) |
33 |
5
|
oien |
|- ( ( ( F supp (/) ) e. _V /\ _E We ( F supp (/) ) ) -> dom G ~~ ( F supp (/) ) ) |
34 |
30 32 33
|
syl2anc |
|- ( ph -> dom G ~~ ( F supp (/) ) ) |
35 |
34
|
adantr |
|- ( ( ph /\ dom G = (/) ) -> dom G ~~ ( F supp (/) ) ) |
36 |
29 35
|
eqbrtrrd |
|- ( ( ph /\ dom G = (/) ) -> (/) ~~ ( F supp (/) ) ) |
37 |
36
|
ensymd |
|- ( ( ph /\ dom G = (/) ) -> ( F supp (/) ) ~~ (/) ) |
38 |
|
en0 |
|- ( ( F supp (/) ) ~~ (/) <-> ( F supp (/) ) = (/) ) |
39 |
37 38
|
sylib |
|- ( ( ph /\ dom G = (/) ) -> ( F supp (/) ) = (/) ) |
40 |
|
ss0b |
|- ( ( F supp (/) ) C_ (/) <-> ( F supp (/) ) = (/) ) |
41 |
39 40
|
sylibr |
|- ( ( ph /\ dom G = (/) ) -> ( F supp (/) ) C_ (/) ) |
42 |
2
|
adantr |
|- ( ( ph /\ dom G = (/) ) -> A e. On ) |
43 |
|
0ex |
|- (/) e. _V |
44 |
43
|
a1i |
|- ( ( ph /\ dom G = (/) ) -> (/) e. _V ) |
45 |
25 41 42 44
|
suppssr |
|- ( ( ( ph /\ dom G = (/) ) /\ x e. ( A \ (/) ) ) -> ( F ` x ) = (/) ) |
46 |
28 45
|
sylan2br |
|- ( ( ( ph /\ dom G = (/) ) /\ x e. A ) -> ( F ` x ) = (/) ) |
47 |
46
|
mpteq2dva |
|- ( ( ph /\ dom G = (/) ) -> ( x e. A |-> ( F ` x ) ) = ( x e. A |-> (/) ) ) |
48 |
26 47
|
eqtrd |
|- ( ( ph /\ dom G = (/) ) -> F = ( x e. A |-> (/) ) ) |
49 |
|
fconstmpt |
|- ( A X. { (/) } ) = ( x e. A |-> (/) ) |
50 |
48 49
|
eqtr4di |
|- ( ( ph /\ dom G = (/) ) -> F = ( A X. { (/) } ) ) |
51 |
50
|
fveq2d |
|- ( ( ph /\ dom G = (/) ) -> ( ( _om CNF A ) ` F ) = ( ( _om CNF A ) ` ( A X. { (/) } ) ) ) |
52 |
4
|
fveq2i |
|- ( ( _om CNF A ) ` F ) = ( ( _om CNF A ) ` ( `' ( _om CNF A ) ` B ) ) |
53 |
|
f1ocnvfv2 |
|- ( ( ( _om CNF A ) : S -1-1-onto-> ( _om ^o A ) /\ B e. ( _om ^o A ) ) -> ( ( _om CNF A ) ` ( `' ( _om CNF A ) ` B ) ) = B ) |
54 |
16 3 53
|
syl2anc |
|- ( ph -> ( ( _om CNF A ) ` ( `' ( _om CNF A ) ` B ) ) = B ) |
55 |
52 54
|
eqtrid |
|- ( ph -> ( ( _om CNF A ) ` F ) = B ) |
56 |
55
|
adantr |
|- ( ( ph /\ dom G = (/) ) -> ( ( _om CNF A ) ` F ) = B ) |
57 |
|
peano1 |
|- (/) e. _om |
58 |
57
|
a1i |
|- ( ph -> (/) e. _om ) |
59 |
1 15 2 58
|
cantnf0 |
|- ( ph -> ( ( _om CNF A ) ` ( A X. { (/) } ) ) = (/) ) |
60 |
59
|
adantr |
|- ( ( ph /\ dom G = (/) ) -> ( ( _om CNF A ) ` ( A X. { (/) } ) ) = (/) ) |
61 |
51 56 60
|
3eqtr3d |
|- ( ( ph /\ dom G = (/) ) -> B = (/) ) |
62 |
13 61
|
mtand |
|- ( ph -> -. dom G = (/) ) |
63 |
|
nnlim |
|- ( dom G e. _om -> -. Lim dom G ) |
64 |
31 63
|
simpl2im |
|- ( ph -> -. Lim dom G ) |
65 |
|
ioran |
|- ( -. ( dom G = (/) \/ Lim dom G ) <-> ( -. dom G = (/) /\ -. Lim dom G ) ) |
66 |
62 64 65
|
sylanbrc |
|- ( ph -> -. ( dom G = (/) \/ Lim dom G ) ) |
67 |
5
|
oicl |
|- Ord dom G |
68 |
|
unizlim |
|- ( Ord dom G -> ( dom G = U. dom G <-> ( dom G = (/) \/ Lim dom G ) ) ) |
69 |
67 68
|
ax-mp |
|- ( dom G = U. dom G <-> ( dom G = (/) \/ Lim dom G ) ) |
70 |
66 69
|
sylnibr |
|- ( ph -> -. dom G = U. dom G ) |
71 |
|
orduniorsuc |
|- ( Ord dom G -> ( dom G = U. dom G \/ dom G = suc U. dom G ) ) |
72 |
67 71
|
mp1i |
|- ( ph -> ( dom G = U. dom G \/ dom G = suc U. dom G ) ) |
73 |
72
|
ord |
|- ( ph -> ( -. dom G = U. dom G -> dom G = suc U. dom G ) ) |
74 |
70 73
|
mpd |
|- ( ph -> dom G = suc U. dom G ) |