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.1 |
⊢ ( 𝜑 → 𝐼 ∈ dom 𝐺 ) |
11 |
|
cnfcom.2 |
⊢ ( 𝜑 → 𝑂 ∈ ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ) |
12 |
|
cnfcom.3 |
⊢ ( 𝜑 → ( 𝑇 ‘ 𝐼 ) : ( 𝐻 ‘ 𝐼 ) –1-1-onto→ 𝑂 ) |
13 |
|
omelon |
⊢ ω ∈ On |
14 |
|
suppssdm |
⊢ ( 𝐹 supp ∅ ) ⊆ dom 𝐹 |
15 |
13
|
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 |
14 24
|
fssdm |
⊢ ( 𝜑 → ( 𝐹 supp ∅ ) ⊆ 𝐴 ) |
26 |
5
|
oif |
⊢ 𝐺 : dom 𝐺 ⟶ ( 𝐹 supp ∅ ) |
27 |
26
|
ffvelrni |
⊢ ( 𝐼 ∈ dom 𝐺 → ( 𝐺 ‘ 𝐼 ) ∈ ( 𝐹 supp ∅ ) ) |
28 |
10 27
|
syl |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐼 ) ∈ ( 𝐹 supp ∅ ) ) |
29 |
25 28
|
sseldd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐼 ) ∈ 𝐴 ) |
30 |
|
onelon |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝐺 ‘ 𝐼 ) ∈ 𝐴 ) → ( 𝐺 ‘ 𝐼 ) ∈ On ) |
31 |
2 29 30
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐼 ) ∈ On ) |
32 |
|
oecl |
⊢ ( ( ω ∈ On ∧ ( 𝐺 ‘ 𝐼 ) ∈ On ) → ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ∈ On ) |
33 |
13 31 32
|
sylancr |
⊢ ( 𝜑 → ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ∈ On ) |
34 |
24 29
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ∈ ω ) |
35 |
|
nnon |
⊢ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ∈ ω → ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ∈ On ) |
36 |
34 35
|
syl |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ∈ On ) |
37 |
|
omcl |
⊢ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ∈ On ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ∈ On ) → ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ∈ On ) |
38 |
33 36 37
|
syl2anc |
⊢ ( 𝜑 → ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ∈ On ) |
39 |
1 15 2 5 21
|
cantnfcl |
⊢ ( 𝜑 → ( E We ( 𝐹 supp ∅ ) ∧ dom 𝐺 ∈ ω ) ) |
40 |
39
|
simprd |
⊢ ( 𝜑 → dom 𝐺 ∈ ω ) |
41 |
|
elnn |
⊢ ( ( 𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω ) → 𝐼 ∈ ω ) |
42 |
10 40 41
|
syl2anc |
⊢ ( 𝜑 → 𝐼 ∈ ω ) |
43 |
6
|
cantnfvalf |
⊢ 𝐻 : ω ⟶ On |
44 |
43
|
ffvelrni |
⊢ ( 𝐼 ∈ ω → ( 𝐻 ‘ 𝐼 ) ∈ On ) |
45 |
42 44
|
syl |
⊢ ( 𝜑 → ( 𝐻 ‘ 𝐼 ) ∈ On ) |
46 |
|
eqid |
⊢ ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) = ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) |
47 |
46
|
oacomf1o |
⊢ ( ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ∈ On ∧ ( 𝐻 ‘ 𝐼 ) ∈ On ) → ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) : ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o ( 𝐻 ‘ 𝐼 ) ) –1-1-onto→ ( ( 𝐻 ‘ 𝐼 ) +o ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) ) |
48 |
38 45 47
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) : ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o ( 𝐻 ‘ 𝐼 ) ) –1-1-onto→ ( ( 𝐻 ‘ 𝐼 ) +o ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) ) |
49 |
7
|
seqomsuc |
⊢ ( 𝐼 ∈ ω → ( 𝑇 ‘ suc 𝐼 ) = ( 𝐼 ( 𝑘 ∈ V , 𝑓 ∈ V ↦ 𝐾 ) ( 𝑇 ‘ 𝐼 ) ) ) |
50 |
42 49
|
syl |
⊢ ( 𝜑 → ( 𝑇 ‘ suc 𝐼 ) = ( 𝐼 ( 𝑘 ∈ V , 𝑓 ∈ V ↦ 𝐾 ) ( 𝑇 ‘ 𝐼 ) ) ) |
51 |
|
nfcv |
⊢ Ⅎ 𝑢 𝐾 |
52 |
|
nfcv |
⊢ Ⅎ 𝑣 𝐾 |
53 |
|
nfcv |
⊢ Ⅎ 𝑘 ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ↦ ( dom 𝑣 +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) ) |
54 |
|
nfcv |
⊢ Ⅎ 𝑓 ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ↦ ( dom 𝑣 +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) ) |
55 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( dom 𝑓 +o 𝑥 ) = ( dom 𝑓 +o 𝑦 ) ) |
56 |
55
|
cbvmptv |
⊢ ( 𝑥 ∈ 𝑀 ↦ ( dom 𝑓 +o 𝑥 ) ) = ( 𝑦 ∈ 𝑀 ↦ ( dom 𝑓 +o 𝑦 ) ) |
57 |
|
simpl |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → 𝑘 = 𝑢 ) |
58 |
57
|
fveq2d |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑢 ) ) |
59 |
58
|
oveq2d |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ( ω ↑o ( 𝐺 ‘ 𝑘 ) ) = ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ) |
60 |
58
|
fveq2d |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) |
61 |
59 60
|
oveq12d |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ( ( ω ↑o ( 𝐺 ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) ) = ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ) |
62 |
8 61
|
eqtrid |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → 𝑀 = ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ) |
63 |
|
simpr |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → 𝑓 = 𝑣 ) |
64 |
63
|
dmeqd |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → dom 𝑓 = dom 𝑣 ) |
65 |
64
|
oveq1d |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ( dom 𝑓 +o 𝑦 ) = ( dom 𝑣 +o 𝑦 ) ) |
66 |
62 65
|
mpteq12dv |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ( 𝑦 ∈ 𝑀 ↦ ( dom 𝑓 +o 𝑦 ) ) = ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ↦ ( dom 𝑣 +o 𝑦 ) ) ) |
67 |
56 66
|
eqtrid |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ( 𝑥 ∈ 𝑀 ↦ ( dom 𝑓 +o 𝑥 ) ) = ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ↦ ( dom 𝑣 +o 𝑦 ) ) ) |
68 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑀 +o 𝑥 ) = ( 𝑀 +o 𝑦 ) ) |
69 |
68
|
cbvmptv |
⊢ ( 𝑥 ∈ dom 𝑓 ↦ ( 𝑀 +o 𝑥 ) ) = ( 𝑦 ∈ dom 𝑓 ↦ ( 𝑀 +o 𝑦 ) ) |
70 |
62
|
oveq1d |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ( 𝑀 +o 𝑦 ) = ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) |
71 |
64 70
|
mpteq12dv |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ( 𝑦 ∈ dom 𝑓 ↦ ( 𝑀 +o 𝑦 ) ) = ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) ) |
72 |
69 71
|
eqtrid |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ( 𝑥 ∈ dom 𝑓 ↦ ( 𝑀 +o 𝑥 ) ) = ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) ) |
73 |
72
|
cnveqd |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ◡ ( 𝑥 ∈ dom 𝑓 ↦ ( 𝑀 +o 𝑥 ) ) = ◡ ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) ) |
74 |
67 73
|
uneq12d |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → ( ( 𝑥 ∈ 𝑀 ↦ ( dom 𝑓 +o 𝑥 ) ) ∪ ◡ ( 𝑥 ∈ dom 𝑓 ↦ ( 𝑀 +o 𝑥 ) ) ) = ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ↦ ( dom 𝑣 +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) ) ) |
75 |
9 74
|
eqtrid |
⊢ ( ( 𝑘 = 𝑢 ∧ 𝑓 = 𝑣 ) → 𝐾 = ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ↦ ( dom 𝑣 +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) ) ) |
76 |
51 52 53 54 75
|
cbvmpo |
⊢ ( 𝑘 ∈ V , 𝑓 ∈ V ↦ 𝐾 ) = ( 𝑢 ∈ V , 𝑣 ∈ V ↦ ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ↦ ( dom 𝑣 +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) ) ) |
77 |
76
|
a1i |
⊢ ( 𝜑 → ( 𝑘 ∈ V , 𝑓 ∈ V ↦ 𝐾 ) = ( 𝑢 ∈ V , 𝑣 ∈ V ↦ ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ↦ ( dom 𝑣 +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) ) ) ) |
78 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → 𝑢 = 𝐼 ) |
79 |
78
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → ( 𝐺 ‘ 𝑢 ) = ( 𝐺 ‘ 𝐼 ) ) |
80 |
79
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) = ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ) |
81 |
79
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) |
82 |
80 81
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) = ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) |
83 |
|
simpr |
⊢ ( ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) → 𝑣 = ( 𝑇 ‘ 𝐼 ) ) |
84 |
83
|
dmeqd |
⊢ ( ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) → dom 𝑣 = dom ( 𝑇 ‘ 𝐼 ) ) |
85 |
|
f1odm |
⊢ ( ( 𝑇 ‘ 𝐼 ) : ( 𝐻 ‘ 𝐼 ) –1-1-onto→ 𝑂 → dom ( 𝑇 ‘ 𝐼 ) = ( 𝐻 ‘ 𝐼 ) ) |
86 |
12 85
|
syl |
⊢ ( 𝜑 → dom ( 𝑇 ‘ 𝐼 ) = ( 𝐻 ‘ 𝐼 ) ) |
87 |
84 86
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → dom 𝑣 = ( 𝐻 ‘ 𝐼 ) ) |
88 |
87
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → ( dom 𝑣 +o 𝑦 ) = ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) |
89 |
82 88
|
mpteq12dv |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ↦ ( dom 𝑣 +o 𝑦 ) ) = ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ) |
90 |
82
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) = ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) |
91 |
87 90
|
mpteq12dv |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) = ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) |
92 |
91
|
cnveqd |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → ◡ ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) = ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) |
93 |
89 92
|
uneq12d |
⊢ ( ( 𝜑 ∧ ( 𝑢 = 𝐼 ∧ 𝑣 = ( 𝑇 ‘ 𝐼 ) ) ) → ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) ↦ ( dom 𝑣 +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ dom 𝑣 ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑢 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑢 ) ) ) +o 𝑦 ) ) ) = ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) ) |
94 |
10
|
elexd |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
95 |
|
fvexd |
⊢ ( 𝜑 → ( 𝑇 ‘ 𝐼 ) ∈ V ) |
96 |
|
ovex |
⊢ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ∈ V |
97 |
96
|
mptex |
⊢ ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ∈ V |
98 |
|
fvex |
⊢ ( 𝐻 ‘ 𝐼 ) ∈ V |
99 |
98
|
mptex |
⊢ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ∈ V |
100 |
99
|
cnvex |
⊢ ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ∈ V |
101 |
97 100
|
unex |
⊢ ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) ∈ V |
102 |
101
|
a1i |
⊢ ( 𝜑 → ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) ∈ V ) |
103 |
77 93 94 95 102
|
ovmpod |
⊢ ( 𝜑 → ( 𝐼 ( 𝑘 ∈ V , 𝑓 ∈ V ↦ 𝐾 ) ( 𝑇 ‘ 𝐼 ) ) = ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) ) |
104 |
50 103
|
eqtrd |
⊢ ( 𝜑 → ( 𝑇 ‘ suc 𝐼 ) = ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) ) |
105 |
104
|
f1oeq1d |
⊢ ( 𝜑 → ( ( 𝑇 ‘ suc 𝐼 ) : ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o ( 𝐻 ‘ 𝐼 ) ) –1-1-onto→ ( ( 𝐻 ‘ 𝐼 ) +o ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) ↔ ( ( 𝑦 ∈ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ↦ ( ( 𝐻 ‘ 𝐼 ) +o 𝑦 ) ) ∪ ◡ ( 𝑦 ∈ ( 𝐻 ‘ 𝐼 ) ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o 𝑦 ) ) ) : ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o ( 𝐻 ‘ 𝐼 ) ) –1-1-onto→ ( ( 𝐻 ‘ 𝐼 ) +o ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) ) ) |
106 |
48 105
|
mpbird |
⊢ ( 𝜑 → ( 𝑇 ‘ suc 𝐼 ) : ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o ( 𝐻 ‘ 𝐼 ) ) –1-1-onto→ ( ( 𝐻 ‘ 𝐼 ) +o ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) ) |
107 |
13
|
a1i |
⊢ ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) → ω ∈ On ) |
108 |
|
simpl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) → 𝐴 ∈ On ) |
109 |
|
simpr |
⊢ ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) → 𝐹 ∈ 𝑆 ) |
110 |
8
|
oveq1i |
⊢ ( 𝑀 +o 𝑧 ) = ( ( ( ω ↑o ( 𝐺 ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) ) +o 𝑧 ) |
111 |
110
|
a1i |
⊢ ( ( 𝑘 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑀 +o 𝑧 ) = ( ( ( ω ↑o ( 𝐺 ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) ) +o 𝑧 ) ) |
112 |
111
|
mpoeq3ia |
⊢ ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( 𝑀 +o 𝑧 ) ) = ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) ) +o 𝑧 ) ) |
113 |
|
eqid |
⊢ ∅ = ∅ |
114 |
|
seqomeq12 |
⊢ ( ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( 𝑀 +o 𝑧 ) ) = ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) ) +o 𝑧 ) ) ∧ ∅ = ∅ ) → seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( 𝑀 +o 𝑧 ) ) , ∅ ) = seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) ) |
115 |
112 113 114
|
mp2an |
⊢ seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( 𝑀 +o 𝑧 ) ) , ∅ ) = seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) |
116 |
6 115
|
eqtri |
⊢ 𝐻 = seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( ω ↑o ( 𝐺 ‘ 𝑘 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) |
117 |
1 107 108 5 109 116
|
cantnfsuc |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) ∧ 𝐼 ∈ ω ) → ( 𝐻 ‘ suc 𝐼 ) = ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o ( 𝐻 ‘ 𝐼 ) ) ) |
118 |
2 21 42 117
|
syl21anc |
⊢ ( 𝜑 → ( 𝐻 ‘ suc 𝐼 ) = ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o ( 𝐻 ‘ 𝐼 ) ) ) |
119 |
118
|
f1oeq2d |
⊢ ( 𝜑 → ( ( 𝑇 ‘ suc 𝐼 ) : ( 𝐻 ‘ suc 𝐼 ) –1-1-onto→ ( ( 𝐻 ‘ 𝐼 ) +o ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) ↔ ( 𝑇 ‘ suc 𝐼 ) : ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) +o ( 𝐻 ‘ 𝐼 ) ) –1-1-onto→ ( ( 𝐻 ‘ 𝐼 ) +o ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) ) ) |
120 |
106 119
|
mpbird |
⊢ ( 𝜑 → ( 𝑇 ‘ suc 𝐼 ) : ( 𝐻 ‘ suc 𝐼 ) –1-1-onto→ ( ( 𝐻 ‘ 𝐼 ) +o ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) ) |
121 |
|
sssucid |
⊢ dom 𝐺 ⊆ suc dom 𝐺 |
122 |
121 10
|
sselid |
⊢ ( 𝜑 → 𝐼 ∈ suc dom 𝐺 ) |
123 |
|
epelg |
⊢ ( 𝐼 ∈ dom 𝐺 → ( 𝑦 E 𝐼 ↔ 𝑦 ∈ 𝐼 ) ) |
124 |
10 123
|
syl |
⊢ ( 𝜑 → ( 𝑦 E 𝐼 ↔ 𝑦 ∈ 𝐼 ) ) |
125 |
124
|
biimpar |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → 𝑦 E 𝐼 ) |
126 |
|
ovexd |
⊢ ( 𝜑 → ( 𝐹 supp ∅ ) ∈ V ) |
127 |
39
|
simpld |
⊢ ( 𝜑 → E We ( 𝐹 supp ∅ ) ) |
128 |
5
|
oiiso |
⊢ ( ( ( 𝐹 supp ∅ ) ∈ V ∧ E We ( 𝐹 supp ∅ ) ) → 𝐺 Isom E , E ( dom 𝐺 , ( 𝐹 supp ∅ ) ) ) |
129 |
126 127 128
|
syl2anc |
⊢ ( 𝜑 → 𝐺 Isom E , E ( dom 𝐺 , ( 𝐹 supp ∅ ) ) ) |
130 |
129
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → 𝐺 Isom E , E ( dom 𝐺 , ( 𝐹 supp ∅ ) ) ) |
131 |
5
|
oicl |
⊢ Ord dom 𝐺 |
132 |
|
ordelss |
⊢ ( ( Ord dom 𝐺 ∧ 𝐼 ∈ dom 𝐺 ) → 𝐼 ⊆ dom 𝐺 ) |
133 |
131 10 132
|
sylancr |
⊢ ( 𝜑 → 𝐼 ⊆ dom 𝐺 ) |
134 |
133
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → 𝑦 ∈ dom 𝐺 ) |
135 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → 𝐼 ∈ dom 𝐺 ) |
136 |
|
isorel |
⊢ ( ( 𝐺 Isom E , E ( dom 𝐺 , ( 𝐹 supp ∅ ) ) ∧ ( 𝑦 ∈ dom 𝐺 ∧ 𝐼 ∈ dom 𝐺 ) ) → ( 𝑦 E 𝐼 ↔ ( 𝐺 ‘ 𝑦 ) E ( 𝐺 ‘ 𝐼 ) ) ) |
137 |
130 134 135 136
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝑦 E 𝐼 ↔ ( 𝐺 ‘ 𝑦 ) E ( 𝐺 ‘ 𝐼 ) ) ) |
138 |
125 137
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑦 ) E ( 𝐺 ‘ 𝐼 ) ) |
139 |
|
fvex |
⊢ ( 𝐺 ‘ 𝐼 ) ∈ V |
140 |
139
|
epeli |
⊢ ( ( 𝐺 ‘ 𝑦 ) E ( 𝐺 ‘ 𝐼 ) ↔ ( 𝐺 ‘ 𝑦 ) ∈ ( 𝐺 ‘ 𝐼 ) ) |
141 |
138 140
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑦 ) ∈ ( 𝐺 ‘ 𝐼 ) ) |
142 |
141
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐼 ( 𝐺 ‘ 𝑦 ) ∈ ( 𝐺 ‘ 𝐼 ) ) |
143 |
|
ffun |
⊢ ( 𝐺 : dom 𝐺 ⟶ ( 𝐹 supp ∅ ) → Fun 𝐺 ) |
144 |
26 143
|
ax-mp |
⊢ Fun 𝐺 |
145 |
|
funimass4 |
⊢ ( ( Fun 𝐺 ∧ 𝐼 ⊆ dom 𝐺 ) → ( ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ↔ ∀ 𝑦 ∈ 𝐼 ( 𝐺 ‘ 𝑦 ) ∈ ( 𝐺 ‘ 𝐼 ) ) ) |
146 |
144 133 145
|
sylancr |
⊢ ( 𝜑 → ( ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ↔ ∀ 𝑦 ∈ 𝐼 ( 𝐺 ‘ 𝑦 ) ∈ ( 𝐺 ‘ 𝐼 ) ) ) |
147 |
142 146
|
mpbird |
⊢ ( 𝜑 → ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ) |
148 |
13
|
a1i |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) ∧ ( 𝐼 ∈ suc dom 𝐺 ∧ ( 𝐺 ‘ 𝐼 ) ∈ On ∧ ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ) ) → ω ∈ On ) |
149 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) ∧ ( 𝐼 ∈ suc dom 𝐺 ∧ ( 𝐺 ‘ 𝐼 ) ∈ On ∧ ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ) ) → 𝐴 ∈ On ) |
150 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) ∧ ( 𝐼 ∈ suc dom 𝐺 ∧ ( 𝐺 ‘ 𝐼 ) ∈ On ∧ ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ) ) → 𝐹 ∈ 𝑆 ) |
151 |
|
peano1 |
⊢ ∅ ∈ ω |
152 |
151
|
a1i |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) ∧ ( 𝐼 ∈ suc dom 𝐺 ∧ ( 𝐺 ‘ 𝐼 ) ∈ On ∧ ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ) ) → ∅ ∈ ω ) |
153 |
|
simpr1 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) ∧ ( 𝐼 ∈ suc dom 𝐺 ∧ ( 𝐺 ‘ 𝐼 ) ∈ On ∧ ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ) ) → 𝐼 ∈ suc dom 𝐺 ) |
154 |
|
simpr2 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) ∧ ( 𝐼 ∈ suc dom 𝐺 ∧ ( 𝐺 ‘ 𝐼 ) ∈ On ∧ ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ) ) → ( 𝐺 ‘ 𝐼 ) ∈ On ) |
155 |
|
simpr3 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) ∧ ( 𝐼 ∈ suc dom 𝐺 ∧ ( 𝐺 ‘ 𝐼 ) ∈ On ∧ ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ) ) → ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ) |
156 |
1 148 149 5 150 116 152 153 154 155
|
cantnflt |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐹 ∈ 𝑆 ) ∧ ( 𝐼 ∈ suc dom 𝐺 ∧ ( 𝐺 ‘ 𝐼 ) ∈ On ∧ ( 𝐺 “ 𝐼 ) ⊆ ( 𝐺 ‘ 𝐼 ) ) ) → ( 𝐻 ‘ 𝐼 ) ∈ ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ) |
157 |
2 21 122 31 147 156
|
syl23anc |
⊢ ( 𝜑 → ( 𝐻 ‘ 𝐼 ) ∈ ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ) |
158 |
24
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
159 |
|
0ex |
⊢ ∅ ∈ V |
160 |
159
|
a1i |
⊢ ( 𝜑 → ∅ ∈ V ) |
161 |
|
elsuppfn |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ On ∧ ∅ ∈ V ) → ( ( 𝐺 ‘ 𝐼 ) ∈ ( 𝐹 supp ∅ ) ↔ ( ( 𝐺 ‘ 𝐼 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ≠ ∅ ) ) ) |
162 |
158 2 160 161
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐺 ‘ 𝐼 ) ∈ ( 𝐹 supp ∅ ) ↔ ( ( 𝐺 ‘ 𝐼 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ≠ ∅ ) ) ) |
163 |
|
simpr |
⊢ ( ( ( 𝐺 ‘ 𝐼 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ≠ ∅ ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ≠ ∅ ) |
164 |
162 163
|
syl6bi |
⊢ ( 𝜑 → ( ( 𝐺 ‘ 𝐼 ) ∈ ( 𝐹 supp ∅ ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ≠ ∅ ) ) |
165 |
28 164
|
mpd |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ≠ ∅ ) |
166 |
|
on0eln0 |
⊢ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ∈ On → ( ∅ ∈ ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ≠ ∅ ) ) |
167 |
36 166
|
syl |
⊢ ( 𝜑 → ( ∅ ∈ ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ≠ ∅ ) ) |
168 |
165 167
|
mpbird |
⊢ ( 𝜑 → ∅ ∈ ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) |
169 |
|
omword1 |
⊢ ( ( ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ∈ On ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ∈ On ) ∧ ∅ ∈ ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) → ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ⊆ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) |
170 |
33 36 168 169
|
syl21anc |
⊢ ( 𝜑 → ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ⊆ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) |
171 |
|
oaabs2 |
⊢ ( ( ( ( 𝐻 ‘ 𝐼 ) ∈ ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ∧ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ∈ On ) ∧ ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ⊆ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) → ( ( 𝐻 ‘ 𝐼 ) +o ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) = ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) |
172 |
157 38 170 171
|
syl21anc |
⊢ ( 𝜑 → ( ( 𝐻 ‘ 𝐼 ) +o ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) = ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) |
173 |
172
|
f1oeq3d |
⊢ ( 𝜑 → ( ( 𝑇 ‘ suc 𝐼 ) : ( 𝐻 ‘ suc 𝐼 ) –1-1-onto→ ( ( 𝐻 ‘ 𝐼 ) +o ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) ↔ ( 𝑇 ‘ suc 𝐼 ) : ( 𝐻 ‘ suc 𝐼 ) –1-1-onto→ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) ) |
174 |
120 173
|
mpbid |
⊢ ( 𝜑 → ( 𝑇 ‘ suc 𝐼 ) : ( 𝐻 ‘ suc 𝐼 ) –1-1-onto→ ( ( ω ↑o ( 𝐺 ‘ 𝐼 ) ) ·o ( 𝐹 ‘ ( 𝐺 ‘ 𝐼 ) ) ) ) |