Step | Hyp | Ref
| Expression |
1 | | elmapi 8285 |
. . . . . . . . . 10
⊢ (𝑜 ∈ (ℕ0
↑𝑚 𝐽) → 𝑜:𝐽⟶ℕ0) |
2 | 1 | adantr 481 |
. . . . . . . . 9
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → 𝑜:𝐽⟶ℕ0) |
3 | | c0ex 10488 |
. . . . . . . . . . 11
⊢ 0 ∈
V |
4 | 3 | fconst 6440 |
. . . . . . . . . 10
⊢ ((ℕ
∖ 𝐽) ×
{0}):(ℕ ∖ 𝐽)⟶{0} |
5 | 4 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → ((ℕ ∖ 𝐽) × {0}):(ℕ ∖ 𝐽)⟶{0}) |
6 | | disjdif 4341 |
. . . . . . . . . 10
⊢ (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅ |
7 | 6 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅) |
8 | | fun 6415 |
. . . . . . . . 9
⊢ (((𝑜:𝐽⟶ℕ0 ∧ ((ℕ
∖ 𝐽) ×
{0}):(ℕ ∖ 𝐽)⟶{0}) ∧ (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0 ∪
{0})) |
9 | 2, 5, 7, 8 | syl21anc 834 |
. . . . . . . 8
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0 ∪
{0})) |
10 | | eulerpart.j |
. . . . . . . . . . 11
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} |
11 | | ssrab2 3983 |
. . . . . . . . . . 11
⊢ {𝑧 ∈ ℕ ∣ ¬ 2
∥ 𝑧} ⊆
ℕ |
12 | 10, 11 | eqsstri 3928 |
. . . . . . . . . 10
⊢ 𝐽 ⊆
ℕ |
13 | | undif 4350 |
. . . . . . . . . 10
⊢ (𝐽 ⊆ ℕ ↔ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ) |
14 | 12, 13 | mpbi 231 |
. . . . . . . . 9
⊢ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ |
15 | | 0nn0 11766 |
. . . . . . . . . . 11
⊢ 0 ∈
ℕ0 |
16 | | snssi 4654 |
. . . . . . . . . . 11
⊢ (0 ∈
ℕ0 → {0} ⊆ ℕ0) |
17 | 15, 16 | ax-mp 5 |
. . . . . . . . . 10
⊢ {0}
⊆ ℕ0 |
18 | | ssequn2 4086 |
. . . . . . . . . 10
⊢ ({0}
⊆ ℕ0 ↔ (ℕ0 ∪ {0}) =
ℕ0) |
19 | 17, 18 | mpbi 231 |
. . . . . . . . 9
⊢
(ℕ0 ∪ {0}) = ℕ0 |
20 | 14, 19 | feq23i 6383 |
. . . . . . . 8
⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0
∪ {0}) ↔ (𝑜 ∪
((ℕ ∖ 𝐽)
× {0})):ℕ⟶ℕ0) |
21 | 9, 20 | sylib 219 |
. . . . . . 7
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) ×
{0})):ℕ⟶ℕ0) |
22 | | nn0ex 11757 |
. . . . . . . 8
⊢
ℕ0 ∈ V |
23 | | nnex 11498 |
. . . . . . . 8
⊢ ℕ
∈ V |
24 | 22, 23 | elmap 8292 |
. . . . . . 7
⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈
(ℕ0 ↑𝑚 ℕ) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) ×
{0})):ℕ⟶ℕ0) |
25 | 21, 24 | sylibr 235 |
. . . . . 6
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0
↑𝑚 ℕ)) |
26 | | cnvun 5884 |
. . . . . . . . 9
⊢ ◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) = (◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0})) |
27 | 26 | imaeq1i 5810 |
. . . . . . . 8
⊢ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0})) “
ℕ) |
28 | | imaundir 5892 |
. . . . . . . 8
⊢ ((◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “
ℕ)) |
29 | 27, 28 | eqtri 2821 |
. . . . . . 7
⊢ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “
ℕ)) |
30 | | vex 3443 |
. . . . . . . . . . 11
⊢ 𝑜 ∈ V |
31 | | cnveq 5637 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑜 → ◡𝑓 = ◡𝑜) |
32 | 31 | imaeq1d 5812 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑜 → (◡𝑓 “ ℕ) = (◡𝑜 “ ℕ)) |
33 | 32 | eleq1d 2869 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝑜 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝑜 “ ℕ) ∈
Fin)) |
34 | | eulerpart.r |
. . . . . . . . . . 11
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈
Fin} |
35 | 30, 33, 34 | elab2 3611 |
. . . . . . . . . 10
⊢ (𝑜 ∈ 𝑅 ↔ (◡𝑜 “ ℕ) ∈
Fin) |
36 | 35 | biimpi 217 |
. . . . . . . . 9
⊢ (𝑜 ∈ 𝑅 → (◡𝑜 “ ℕ) ∈
Fin) |
37 | 36 | adantl 482 |
. . . . . . . 8
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡𝑜 “ ℕ) ∈
Fin) |
38 | | cnvxp 5897 |
. . . . . . . . . . . . . 14
⊢ ◡((ℕ ∖ 𝐽) × {0}) = ({0} × (ℕ
∖ 𝐽)) |
39 | 38 | dmeqi 5666 |
. . . . . . . . . . . . 13
⊢ dom ◡((ℕ ∖ 𝐽) × {0}) = dom ({0} × (ℕ
∖ 𝐽)) |
40 | | 2nn 11564 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℕ |
41 | | 2z 11868 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℤ |
42 | | iddvds 15460 |
. . . . . . . . . . . . . . . . 17
⊢ (2 ∈
ℤ → 2 ∥ 2) |
43 | 41, 42 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∥
2 |
44 | | breq2 4972 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = 2 → (2 ∥ 𝑧 ↔ 2 ∥
2)) |
45 | 44 | notbid 319 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 2 → (¬ 2 ∥
𝑧 ↔ ¬ 2 ∥
2)) |
46 | 45, 10 | elrab2 3624 |
. . . . . . . . . . . . . . . . 17
⊢ (2 ∈
𝐽 ↔ (2 ∈ ℕ
∧ ¬ 2 ∥ 2)) |
47 | 46 | simprbi 497 |
. . . . . . . . . . . . . . . 16
⊢ (2 ∈
𝐽 → ¬ 2 ∥
2) |
48 | 43, 47 | mt2 201 |
. . . . . . . . . . . . . . 15
⊢ ¬ 2
∈ 𝐽 |
49 | | eldif 3875 |
. . . . . . . . . . . . . . 15
⊢ (2 ∈
(ℕ ∖ 𝐽) ↔
(2 ∈ ℕ ∧ ¬ 2 ∈ 𝐽)) |
50 | 40, 48, 49 | mpbir2an 707 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
(ℕ ∖ 𝐽) |
51 | | ne0i 4226 |
. . . . . . . . . . . . . 14
⊢ (2 ∈
(ℕ ∖ 𝐽) →
(ℕ ∖ 𝐽) ≠
∅) |
52 | | dmxp 5688 |
. . . . . . . . . . . . . 14
⊢ ((ℕ
∖ 𝐽) ≠ ∅
→ dom ({0} × (ℕ ∖ 𝐽)) = {0}) |
53 | 50, 51, 52 | mp2b 10 |
. . . . . . . . . . . . 13
⊢ dom ({0}
× (ℕ ∖ 𝐽)) = {0} |
54 | 39, 53 | eqtri 2821 |
. . . . . . . . . . . 12
⊢ dom ◡((ℕ ∖ 𝐽) × {0}) = {0} |
55 | 54 | ineq1i 4111 |
. . . . . . . . . . 11
⊢ (dom
◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ({0} ∩
ℕ) |
56 | | incom 4105 |
. . . . . . . . . . 11
⊢ (ℕ
∩ {0}) = ({0} ∩ ℕ) |
57 | | 0nnn 11527 |
. . . . . . . . . . . 12
⊢ ¬ 0
∈ ℕ |
58 | | disjsn 4560 |
. . . . . . . . . . . 12
⊢ ((ℕ
∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ) |
59 | 57, 58 | mpbir 232 |
. . . . . . . . . . 11
⊢ (ℕ
∩ {0}) = ∅ |
60 | 55, 56, 59 | 3eqtr2i 2827 |
. . . . . . . . . 10
⊢ (dom
◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) =
∅ |
61 | | imadisj 5831 |
. . . . . . . . . 10
⊢ ((◡((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅
↔ (dom ◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) =
∅) |
62 | 60, 61 | mpbir 232 |
. . . . . . . . 9
⊢ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) =
∅ |
63 | | 0fin 8599 |
. . . . . . . . 9
⊢ ∅
∈ Fin |
64 | 62, 63 | eqeltri 2881 |
. . . . . . . 8
⊢ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈
Fin |
65 | | unfi 8638 |
. . . . . . . 8
⊢ (((◡𝑜 “ ℕ) ∈ Fin ∧ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈ Fin)
→ ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ)) ∈
Fin) |
66 | 37, 64, 65 | sylancl 586 |
. . . . . . 7
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ)) ∈
Fin) |
67 | 29, 66 | syl5eqel 2889 |
. . . . . 6
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ∈
Fin) |
68 | | cnvimass 5832 |
. . . . . . . . 9
⊢ (◡𝑜 “ ℕ) ⊆ dom 𝑜 |
69 | 68, 2 | fssdm 6405 |
. . . . . . . 8
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡𝑜 “ ℕ) ⊆ 𝐽) |
70 | | 0ss 4276 |
. . . . . . . . . 10
⊢ ∅
⊆ 𝐽 |
71 | 62, 70 | eqsstri 3928 |
. . . . . . . . 9
⊢ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆
𝐽 |
72 | 71 | a1i 11 |
. . . . . . . 8
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ⊆
𝐽) |
73 | 69, 72 | unssd 4089 |
. . . . . . 7
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ)) ⊆
𝐽) |
74 | 29, 73 | eqsstrid 3942 |
. . . . . 6
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ⊆
𝐽) |
75 | | eulerpart.p |
. . . . . . 7
⊢ 𝑃 = {𝑓 ∈ (ℕ0
↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧
Σ𝑘 ∈ ℕ
((𝑓‘𝑘) · 𝑘) = 𝑁)} |
76 | | eulerpart.o |
. . . . . . 7
⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} |
77 | | eulerpart.d |
. . . . . . 7
⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} |
78 | | eulerpart.f |
. . . . . . 7
⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦
((2↑𝑦) · 𝑥)) |
79 | | eulerpart.h |
. . . . . . 7
⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩
Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} |
80 | | eulerpart.m |
. . . . . . 7
⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) |
81 | | eulerpart.t |
. . . . . . 7
⊢ 𝑇 = {𝑓 ∈ (ℕ0
↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} |
82 | 75, 76, 77, 10, 78, 79, 80, 34, 81 | eulerpartlemt0 31240 |
. . . . . 6
⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇 ∩ 𝑅) ↔ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0
↑𝑚 ℕ) ∧ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ∈ Fin
∧ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ⊆
𝐽)) |
83 | 25, 67, 74, 82 | syl3anbrc 1336 |
. . . . 5
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇 ∩ 𝑅)) |
84 | | resundir 5756 |
. . . . . 6
⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽) = ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) |
85 | | ffn 6389 |
. . . . . . . 8
⊢ (𝑜:𝐽⟶ℕ0 → 𝑜 Fn 𝐽) |
86 | | fnresdm 6343 |
. . . . . . . . 9
⊢ (𝑜 Fn 𝐽 → (𝑜 ↾ 𝐽) = 𝑜) |
87 | | incom 4105 |
. . . . . . . . . . . 12
⊢ ((ℕ
∖ 𝐽) ∩ 𝐽) = (𝐽 ∩ (ℕ ∖ 𝐽)) |
88 | 87, 6 | eqtri 2821 |
. . . . . . . . . . 11
⊢ ((ℕ
∖ 𝐽) ∩ 𝐽) = ∅ |
89 | | fnconstg 6442 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℕ0 → ((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽)) |
90 | | fnresdisj 6344 |
. . . . . . . . . . . 12
⊢
(((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽) → (((ℕ ∖
𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅)) |
91 | 15, 89, 90 | mp2b 10 |
. . . . . . . . . . 11
⊢
(((ℕ ∖ 𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅) |
92 | 88, 91 | mpbi 231 |
. . . . . . . . . 10
⊢
(((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅ |
93 | 92 | a1i 11 |
. . . . . . . . 9
⊢ (𝑜 Fn 𝐽 → (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅) |
94 | 86, 93 | uneq12d 4067 |
. . . . . . . 8
⊢ (𝑜 Fn 𝐽 → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅)) |
95 | 2, 85, 94 | 3syl 18 |
. . . . . . 7
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅)) |
96 | | un0 4270 |
. . . . . . 7
⊢ (𝑜 ∪ ∅) = 𝑜 |
97 | 95, 96 | syl6eq 2849 |
. . . . . 6
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = 𝑜) |
98 | 84, 97 | syl5req 2846 |
. . . . 5
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽)) |
99 | | reseq1 5735 |
. . . . . 6
⊢ (𝑚 = (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) → (𝑚 ↾ 𝐽) = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽)) |
100 | 99 | rspceeqv 3579 |
. . . . 5
⊢ (((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽)) → ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)) |
101 | 83, 98, 100 | syl2anc 584 |
. . . 4
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) → ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)) |
102 | | simpr 485 |
. . . . . . 7
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 = (𝑚 ↾ 𝐽)) |
103 | | simpl 483 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑚 ∈ (𝑇 ∩ 𝑅)) |
104 | 75, 76, 77, 10, 78, 79, 80, 34, 81 | eulerpartlemt0 31240 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ (𝑇 ∩ 𝑅) ↔ (𝑚 ∈ (ℕ0
↑𝑚 ℕ) ∧ (◡𝑚 “ ℕ) ∈ Fin ∧ (◡𝑚 “ ℕ) ⊆ 𝐽)) |
105 | 103, 104 | sylib 219 |
. . . . . . . . . . 11
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ∈ (ℕ0
↑𝑚 ℕ) ∧ (◡𝑚 “ ℕ) ∈ Fin ∧ (◡𝑚 “ ℕ) ⊆ 𝐽)) |
106 | 105 | simp1d 1135 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑚 ∈ (ℕ0
↑𝑚 ℕ)) |
107 | 22, 23 | elmap 8292 |
. . . . . . . . . 10
⊢ (𝑚 ∈ (ℕ0
↑𝑚 ℕ) ↔ 𝑚:ℕ⟶ℕ0) |
108 | 106, 107 | sylib 219 |
. . . . . . . . 9
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑚:ℕ⟶ℕ0) |
109 | | fssres 6419 |
. . . . . . . . 9
⊢ ((𝑚:ℕ⟶ℕ0 ∧
𝐽 ⊆ ℕ) →
(𝑚 ↾ 𝐽):𝐽⟶ℕ0) |
110 | 108, 12, 109 | sylancl 586 |
. . . . . . . 8
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽):𝐽⟶ℕ0) |
111 | 10, 23 | rabex2 5135 |
. . . . . . . . 9
⊢ 𝐽 ∈ V |
112 | 22, 111 | elmap 8292 |
. . . . . . . 8
⊢ ((𝑚 ↾ 𝐽) ∈ (ℕ0
↑𝑚 𝐽) ↔ (𝑚 ↾ 𝐽):𝐽⟶ℕ0) |
113 | 110, 112 | sylibr 235 |
. . . . . . 7
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽) ∈ (ℕ0
↑𝑚 𝐽)) |
114 | 102, 113 | eqeltrd 2885 |
. . . . . 6
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 ∈ (ℕ0
↑𝑚 𝐽)) |
115 | | ffun 6392 |
. . . . . . . . . 10
⊢ (𝑚:ℕ⟶ℕ0 →
Fun 𝑚) |
116 | | respreima 6708 |
. . . . . . . . . 10
⊢ (Fun
𝑚 → (◡(𝑚 ↾ 𝐽) “ ℕ) = ((◡𝑚 “ ℕ) ∩ 𝐽)) |
117 | 108, 115,
116 | 3syl 18 |
. . . . . . . . 9
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡(𝑚 ↾ 𝐽) “ ℕ) = ((◡𝑚 “ ℕ) ∩ 𝐽)) |
118 | 105 | simp2d 1136 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡𝑚 “ ℕ) ∈
Fin) |
119 | | infi 8595 |
. . . . . . . . . 10
⊢ ((◡𝑚 “ ℕ) ∈ Fin → ((◡𝑚 “ ℕ) ∩ 𝐽) ∈ Fin) |
120 | 118, 119 | syl 17 |
. . . . . . . . 9
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → ((◡𝑚 “ ℕ) ∩ 𝐽) ∈ Fin) |
121 | 117, 120 | eqeltrd 2885 |
. . . . . . . 8
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡(𝑚 ↾ 𝐽) “ ℕ) ∈
Fin) |
122 | | vex 3443 |
. . . . . . . . . 10
⊢ 𝑚 ∈ V |
123 | 122 | resex 5787 |
. . . . . . . . 9
⊢ (𝑚 ↾ 𝐽) ∈ V |
124 | | cnveq 5637 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑚 ↾ 𝐽) → ◡𝑓 = ◡(𝑚 ↾ 𝐽)) |
125 | 124 | imaeq1d 5812 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑚 ↾ 𝐽) → (◡𝑓 “ ℕ) = (◡(𝑚 ↾ 𝐽) “ ℕ)) |
126 | 125 | eleq1d 2869 |
. . . . . . . . 9
⊢ (𝑓 = (𝑚 ↾ 𝐽) → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡(𝑚 ↾ 𝐽) “ ℕ) ∈
Fin)) |
127 | 123, 126,
34 | elab2 3611 |
. . . . . . . 8
⊢ ((𝑚 ↾ 𝐽) ∈ 𝑅 ↔ (◡(𝑚 ↾ 𝐽) “ ℕ) ∈
Fin) |
128 | 121, 127 | sylibr 235 |
. . . . . . 7
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽) ∈ 𝑅) |
129 | 102, 128 | eqeltrd 2885 |
. . . . . 6
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 ∈ 𝑅) |
130 | 114, 129 | jca 512 |
. . . . 5
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅)) |
131 | 130 | rexlimiva 3246 |
. . . 4
⊢
(∃𝑚 ∈
(𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽) → (𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅)) |
132 | 101, 131 | impbii 210 |
. . 3
⊢ ((𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅) ↔ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)) |
133 | 132 | abbii 2863 |
. 2
⊢ {𝑜 ∣ (𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅)} = {𝑜 ∣ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)} |
134 | | df-in 3872 |
. 2
⊢
((ℕ0 ↑𝑚 𝐽) ∩ 𝑅) = {𝑜 ∣ (𝑜 ∈ (ℕ0
↑𝑚 𝐽) ∧ 𝑜 ∈ 𝑅)} |
135 | | eqid 2797 |
. . 3
⊢ (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) = (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) |
136 | 135 | rnmpt 5716 |
. 2
⊢ ran
(𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) = {𝑜 ∣ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)} |
137 | 133, 134,
136 | 3eqtr4i 2831 |
1
⊢
((ℕ0 ↑𝑚 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) |