Step |
Hyp |
Ref |
Expression |
1 |
|
cnfcom.s |
⊢ 𝑆 = dom ( ω CNF 𝐴 ) |
2 |
|
cnfcom.a |
⊢ ( 𝜑 → 𝐴 ∈ On ) |
3 |
|
cnfcom.b |
⊢ ( 𝜑 → 𝐵 ∈ ( ω ↑o 𝐴 ) ) |
4 |
|
cnfcom.f |
⊢ 𝐹 = ( ◡ ( ω CNF 𝐴 ) ‘ 𝐵 ) |
5 |
|
cnfcom.g |
⊢ 𝐺 = OrdIso ( E , ( 𝐹 supp ∅ ) ) |
6 |
|
cnfcom.h |
⊢ 𝐻 = seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( 𝑀 +o 𝑧 ) ) , ∅ ) |
7 |
|
cnfcom.t |
⊢ 𝑇 = seqω ( ( 𝑘 ∈ V , 𝑓 ∈ V ↦ 𝐾 ) , ∅ ) |
8 |
|
cnfcom.m |
⊢ 𝑀 = ( ( ω ↑o ( 𝐺 ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) ) |
9 |
|
cnfcom.k |
⊢ 𝐾 = ( ( 𝑥 ∈ 𝑀 ↦ ( dom 𝑓 +o 𝑥 ) ) ∪ ◡ ( 𝑥 ∈ dom 𝑓 ↦ ( 𝑀 +o 𝑥 ) ) ) |
10 |
|
cnfcom.w |
⊢ 𝑊 = ( 𝐺 ‘ ∪ dom 𝐺 ) |
11 |
|
cnfcom2.1 |
⊢ ( 𝜑 → ∅ ∈ 𝐵 ) |
12 |
|
n0i |
⊢ ( ∅ ∈ 𝐵 → ¬ 𝐵 = ∅ ) |
13 |
11 12
|
syl |
⊢ ( 𝜑 → ¬ 𝐵 = ∅ ) |
14 |
|
omelon |
⊢ ω ∈ On |
15 |
14
|
a1i |
⊢ ( 𝜑 → ω ∈ On ) |
16 |
1 15 2
|
cantnff1o |
⊢ ( 𝜑 → ( ω CNF 𝐴 ) : 𝑆 –1-1-onto→ ( ω ↑o 𝐴 ) ) |
17 |
|
f1ocnv |
⊢ ( ( ω CNF 𝐴 ) : 𝑆 –1-1-onto→ ( ω ↑o 𝐴 ) → ◡ ( ω CNF 𝐴 ) : ( ω ↑o 𝐴 ) –1-1-onto→ 𝑆 ) |
18 |
|
f1of |
⊢ ( ◡ ( ω CNF 𝐴 ) : ( ω ↑o 𝐴 ) –1-1-onto→ 𝑆 → ◡ ( ω CNF 𝐴 ) : ( ω ↑o 𝐴 ) ⟶ 𝑆 ) |
19 |
16 17 18
|
3syl |
⊢ ( 𝜑 → ◡ ( ω CNF 𝐴 ) : ( ω ↑o 𝐴 ) ⟶ 𝑆 ) |
20 |
19 3
|
ffvelrnd |
⊢ ( 𝜑 → ( ◡ ( ω CNF 𝐴 ) ‘ 𝐵 ) ∈ 𝑆 ) |
21 |
4 20
|
eqeltrid |
⊢ ( 𝜑 → 𝐹 ∈ 𝑆 ) |
22 |
1 15 2
|
cantnfs |
⊢ ( 𝜑 → ( 𝐹 ∈ 𝑆 ↔ ( 𝐹 : 𝐴 ⟶ ω ∧ 𝐹 finSupp ∅ ) ) ) |
23 |
21 22
|
mpbid |
⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ ω ∧ 𝐹 finSupp ∅ ) ) |
24 |
23
|
simpld |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ω ) |
25 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → 𝐹 : 𝐴 ⟶ ω ) |
26 |
25
|
feqmptd |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
27 |
|
dif0 |
⊢ ( 𝐴 ∖ ∅ ) = 𝐴 |
28 |
27
|
eleq2i |
⊢ ( 𝑥 ∈ ( 𝐴 ∖ ∅ ) ↔ 𝑥 ∈ 𝐴 ) |
29 |
|
simpr |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → dom 𝐺 = ∅ ) |
30 |
|
ovexd |
⊢ ( 𝜑 → ( 𝐹 supp ∅ ) ∈ V ) |
31 |
1 15 2 5 21
|
cantnfcl |
⊢ ( 𝜑 → ( E We ( 𝐹 supp ∅ ) ∧ dom 𝐺 ∈ ω ) ) |
32 |
31
|
simpld |
⊢ ( 𝜑 → E We ( 𝐹 supp ∅ ) ) |
33 |
5
|
oien |
⊢ ( ( ( 𝐹 supp ∅ ) ∈ V ∧ E We ( 𝐹 supp ∅ ) ) → dom 𝐺 ≈ ( 𝐹 supp ∅ ) ) |
34 |
30 32 33
|
syl2anc |
⊢ ( 𝜑 → dom 𝐺 ≈ ( 𝐹 supp ∅ ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → dom 𝐺 ≈ ( 𝐹 supp ∅ ) ) |
36 |
29 35
|
eqbrtrrd |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → ∅ ≈ ( 𝐹 supp ∅ ) ) |
37 |
36
|
ensymd |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → ( 𝐹 supp ∅ ) ≈ ∅ ) |
38 |
|
en0 |
⊢ ( ( 𝐹 supp ∅ ) ≈ ∅ ↔ ( 𝐹 supp ∅ ) = ∅ ) |
39 |
37 38
|
sylib |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → ( 𝐹 supp ∅ ) = ∅ ) |
40 |
|
ss0b |
⊢ ( ( 𝐹 supp ∅ ) ⊆ ∅ ↔ ( 𝐹 supp ∅ ) = ∅ ) |
41 |
39 40
|
sylibr |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → ( 𝐹 supp ∅ ) ⊆ ∅ ) |
42 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → 𝐴 ∈ On ) |
43 |
|
0ex |
⊢ ∅ ∈ V |
44 |
43
|
a1i |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → ∅ ∈ V ) |
45 |
25 41 42 44
|
suppssr |
⊢ ( ( ( 𝜑 ∧ dom 𝐺 = ∅ ) ∧ 𝑥 ∈ ( 𝐴 ∖ ∅ ) ) → ( 𝐹 ‘ 𝑥 ) = ∅ ) |
46 |
28 45
|
sylan2br |
⊢ ( ( ( 𝜑 ∧ dom 𝐺 = ∅ ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) = ∅ ) |
47 |
46
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) = ( 𝑥 ∈ 𝐴 ↦ ∅ ) ) |
48 |
26 47
|
eqtrd |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ∅ ) ) |
49 |
|
fconstmpt |
⊢ ( 𝐴 × { ∅ } ) = ( 𝑥 ∈ 𝐴 ↦ ∅ ) |
50 |
48 49
|
eqtr4di |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → 𝐹 = ( 𝐴 × { ∅ } ) ) |
51 |
50
|
fveq2d |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → ( ( ω CNF 𝐴 ) ‘ 𝐹 ) = ( ( ω CNF 𝐴 ) ‘ ( 𝐴 × { ∅ } ) ) ) |
52 |
4
|
fveq2i |
⊢ ( ( ω CNF 𝐴 ) ‘ 𝐹 ) = ( ( ω CNF 𝐴 ) ‘ ( ◡ ( ω CNF 𝐴 ) ‘ 𝐵 ) ) |
53 |
|
f1ocnvfv2 |
⊢ ( ( ( ω CNF 𝐴 ) : 𝑆 –1-1-onto→ ( ω ↑o 𝐴 ) ∧ 𝐵 ∈ ( ω ↑o 𝐴 ) ) → ( ( ω CNF 𝐴 ) ‘ ( ◡ ( ω CNF 𝐴 ) ‘ 𝐵 ) ) = 𝐵 ) |
54 |
16 3 53
|
syl2anc |
⊢ ( 𝜑 → ( ( ω CNF 𝐴 ) ‘ ( ◡ ( ω CNF 𝐴 ) ‘ 𝐵 ) ) = 𝐵 ) |
55 |
52 54
|
eqtrid |
⊢ ( 𝜑 → ( ( ω CNF 𝐴 ) ‘ 𝐹 ) = 𝐵 ) |
56 |
55
|
adantr |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → ( ( ω CNF 𝐴 ) ‘ 𝐹 ) = 𝐵 ) |
57 |
|
peano1 |
⊢ ∅ ∈ ω |
58 |
57
|
a1i |
⊢ ( 𝜑 → ∅ ∈ ω ) |
59 |
1 15 2 58
|
cantnf0 |
⊢ ( 𝜑 → ( ( ω CNF 𝐴 ) ‘ ( 𝐴 × { ∅ } ) ) = ∅ ) |
60 |
59
|
adantr |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → ( ( ω CNF 𝐴 ) ‘ ( 𝐴 × { ∅ } ) ) = ∅ ) |
61 |
51 56 60
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ dom 𝐺 = ∅ ) → 𝐵 = ∅ ) |
62 |
13 61
|
mtand |
⊢ ( 𝜑 → ¬ dom 𝐺 = ∅ ) |
63 |
|
nnlim |
⊢ ( dom 𝐺 ∈ ω → ¬ Lim dom 𝐺 ) |
64 |
31 63
|
simpl2im |
⊢ ( 𝜑 → ¬ Lim dom 𝐺 ) |
65 |
|
ioran |
⊢ ( ¬ ( dom 𝐺 = ∅ ∨ Lim dom 𝐺 ) ↔ ( ¬ dom 𝐺 = ∅ ∧ ¬ Lim dom 𝐺 ) ) |
66 |
62 64 65
|
sylanbrc |
⊢ ( 𝜑 → ¬ ( dom 𝐺 = ∅ ∨ Lim dom 𝐺 ) ) |
67 |
5
|
oicl |
⊢ Ord dom 𝐺 |
68 |
|
unizlim |
⊢ ( Ord dom 𝐺 → ( dom 𝐺 = ∪ dom 𝐺 ↔ ( dom 𝐺 = ∅ ∨ Lim dom 𝐺 ) ) ) |
69 |
67 68
|
ax-mp |
⊢ ( dom 𝐺 = ∪ dom 𝐺 ↔ ( dom 𝐺 = ∅ ∨ Lim dom 𝐺 ) ) |
70 |
66 69
|
sylnibr |
⊢ ( 𝜑 → ¬ dom 𝐺 = ∪ dom 𝐺 ) |
71 |
|
orduniorsuc |
⊢ ( Ord dom 𝐺 → ( dom 𝐺 = ∪ dom 𝐺 ∨ dom 𝐺 = suc ∪ dom 𝐺 ) ) |
72 |
67 71
|
mp1i |
⊢ ( 𝜑 → ( dom 𝐺 = ∪ dom 𝐺 ∨ dom 𝐺 = suc ∪ dom 𝐺 ) ) |
73 |
72
|
ord |
⊢ ( 𝜑 → ( ¬ dom 𝐺 = ∪ dom 𝐺 → dom 𝐺 = suc ∪ dom 𝐺 ) ) |
74 |
70 73
|
mpd |
⊢ ( 𝜑 → dom 𝐺 = suc ∪ dom 𝐺 ) |