| Step | Hyp | Ref
| Expression |
| 1 | | elmapi 8889 |
. . . . . . . . . 10
⊢ (𝑜 ∈ (ℕ0
↑m 𝐽)
→ 𝑜:𝐽⟶ℕ0) |
| 2 | 1 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → 𝑜:𝐽⟶ℕ0) |
| 3 | | c0ex 11255 |
. . . . . . . . . . 11
⊢ 0 ∈
V |
| 4 | 3 | fconst 6794 |
. . . . . . . . . 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 4472 |
. . . . . . . . . 10
⊢ (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅ |
| 7 | 6 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → (𝐽 ∩ (ℕ ∖ 𝐽)) = ∅) |
| 8 | | fun 6770 |
. . . . . . . . 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 838 |
. . . . . . . 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 4080 |
. . . . . . . . . . 11
⊢ {𝑧 ∈ ℕ ∣ ¬ 2
∥ 𝑧} ⊆
ℕ |
| 12 | 10, 11 | eqsstri 4030 |
. . . . . . . . . 10
⊢ 𝐽 ⊆
ℕ |
| 13 | | undif 4482 |
. . . . . . . . . 10
⊢ (𝐽 ⊆ ℕ ↔ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ) |
| 14 | 12, 13 | mpbi 230 |
. . . . . . . . 9
⊢ (𝐽 ∪ (ℕ ∖ 𝐽)) = ℕ |
| 15 | | 0nn0 12541 |
. . . . . . . . . . 11
⊢ 0 ∈
ℕ0 |
| 16 | | snssi 4808 |
. . . . . . . . . . 11
⊢ (0 ∈
ℕ0 → {0} ⊆ ℕ0) |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . . 10
⊢ {0}
⊆ ℕ0 |
| 18 | | ssequn2 4189 |
. . . . . . . . . 10
⊢ ({0}
⊆ ℕ0 ↔ (ℕ0 ∪ {0}) =
ℕ0) |
| 19 | 17, 18 | mpbi 230 |
. . . . . . . . 9
⊢
(ℕ0 ∪ {0}) = ℕ0 |
| 20 | 14, 19 | feq23i 6730 |
. . . . . . . 8
⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})):(𝐽 ∪ (ℕ ∖ 𝐽))⟶(ℕ0
∪ {0}) ↔ (𝑜 ∪
((ℕ ∖ 𝐽)
× {0})):ℕ⟶ℕ0) |
| 21 | 9, 20 | sylib 218 |
. . . . . . 7
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) ×
{0})):ℕ⟶ℕ0) |
| 22 | | nn0ex 12532 |
. . . . . . . 8
⊢
ℕ0 ∈ V |
| 23 | | nnex 12272 |
. . . . . . . 8
⊢ ℕ
∈ V |
| 24 | 22, 23 | elmap 8911 |
. . . . . . 7
⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈
(ℕ0 ↑m ℕ) ↔ (𝑜 ∪ ((ℕ ∖ 𝐽) ×
{0})):ℕ⟶ℕ0) |
| 25 | 21, 24 | sylibr 234 |
. . . . . 6
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (ℕ0
↑m ℕ)) |
| 26 | | cnvun 6162 |
. . . . . . . . 9
⊢ ◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) = (◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0})) |
| 27 | 26 | imaeq1i 6075 |
. . . . . . . 8
⊢ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0})) “
ℕ) |
| 28 | | imaundir 6170 |
. . . . . . . 8
⊢ ((◡𝑜 ∪ ◡((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “
ℕ)) |
| 29 | 27, 28 | eqtri 2765 |
. . . . . . 7
⊢ (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) = ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “
ℕ)) |
| 30 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑜 ∈ V |
| 31 | | cnveq 5884 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑜 → ◡𝑓 = ◡𝑜) |
| 32 | 31 | imaeq1d 6077 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑜 → (◡𝑓 “ ℕ) = (◡𝑜 “ ℕ)) |
| 33 | 32 | eleq1d 2826 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝑜 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝑜 “ ℕ) ∈
Fin)) |
| 34 | | eulerpart.r |
. . . . . . . . . . 11
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈
Fin} |
| 35 | 30, 33, 34 | elab2 3682 |
. . . . . . . . . 10
⊢ (𝑜 ∈ 𝑅 ↔ (◡𝑜 “ ℕ) ∈
Fin) |
| 36 | 35 | biimpi 216 |
. . . . . . . . 9
⊢ (𝑜 ∈ 𝑅 → (◡𝑜 “ ℕ) ∈
Fin) |
| 37 | 36 | adantl 481 |
. . . . . . . 8
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → (◡𝑜 “ ℕ) ∈
Fin) |
| 38 | | cnvxp 6177 |
. . . . . . . . . . . . . 14
⊢ ◡((ℕ ∖ 𝐽) × {0}) = ({0} × (ℕ
∖ 𝐽)) |
| 39 | 38 | dmeqi 5915 |
. . . . . . . . . . . . 13
⊢ dom ◡((ℕ ∖ 𝐽) × {0}) = dom ({0} × (ℕ
∖ 𝐽)) |
| 40 | | 2nn 12339 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℕ |
| 41 | | 2z 12649 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℤ |
| 42 | | iddvds 16307 |
. . . . . . . . . . . . . . . . 17
⊢ (2 ∈
ℤ → 2 ∥ 2) |
| 43 | 41, 42 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∥
2 |
| 44 | | breq2 5147 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = 2 → (2 ∥ 𝑧 ↔ 2 ∥
2)) |
| 45 | 44 | notbid 318 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 2 → (¬ 2 ∥
𝑧 ↔ ¬ 2 ∥
2)) |
| 46 | 45, 10 | elrab2 3695 |
. . . . . . . . . . . . . . . . 17
⊢ (2 ∈
𝐽 ↔ (2 ∈ ℕ
∧ ¬ 2 ∥ 2)) |
| 47 | 46 | simprbi 496 |
. . . . . . . . . . . . . . . 16
⊢ (2 ∈
𝐽 → ¬ 2 ∥
2) |
| 48 | 43, 47 | mt2 200 |
. . . . . . . . . . . . . . 15
⊢ ¬ 2
∈ 𝐽 |
| 49 | | eldif 3961 |
. . . . . . . . . . . . . . 15
⊢ (2 ∈
(ℕ ∖ 𝐽) ↔
(2 ∈ ℕ ∧ ¬ 2 ∈ 𝐽)) |
| 50 | 40, 48, 49 | mpbir2an 711 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
(ℕ ∖ 𝐽) |
| 51 | | ne0i 4341 |
. . . . . . . . . . . . . 14
⊢ (2 ∈
(ℕ ∖ 𝐽) →
(ℕ ∖ 𝐽) ≠
∅) |
| 52 | | dmxp 5939 |
. . . . . . . . . . . . . 14
⊢ ((ℕ
∖ 𝐽) ≠ ∅
→ dom ({0} × (ℕ ∖ 𝐽)) = {0}) |
| 53 | 50, 51, 52 | mp2b 10 |
. . . . . . . . . . . . 13
⊢ dom ({0}
× (ℕ ∖ 𝐽)) = {0} |
| 54 | 39, 53 | eqtri 2765 |
. . . . . . . . . . . 12
⊢ dom ◡((ℕ ∖ 𝐽) × {0}) = {0} |
| 55 | 54 | ineq1i 4216 |
. . . . . . . . . . 11
⊢ (dom
◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) = ({0} ∩
ℕ) |
| 56 | | incom 4209 |
. . . . . . . . . . 11
⊢ (ℕ
∩ {0}) = ({0} ∩ ℕ) |
| 57 | | 0nnn 12302 |
. . . . . . . . . . . 12
⊢ ¬ 0
∈ ℕ |
| 58 | | disjsn 4711 |
. . . . . . . . . . . 12
⊢ ((ℕ
∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ) |
| 59 | 57, 58 | mpbir 231 |
. . . . . . . . . . 11
⊢ (ℕ
∩ {0}) = ∅ |
| 60 | 55, 56, 59 | 3eqtr2i 2771 |
. . . . . . . . . 10
⊢ (dom
◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) =
∅ |
| 61 | | imadisj 6098 |
. . . . . . . . . 10
⊢ ((◡((ℕ ∖ 𝐽) × {0}) “ ℕ) = ∅
↔ (dom ◡((ℕ ∖ 𝐽) × {0}) ∩ ℕ) =
∅) |
| 62 | 60, 61 | mpbir 231 |
. . . . . . . . 9
⊢ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) =
∅ |
| 63 | | 0fi 9082 |
. . . . . . . . 9
⊢ ∅
∈ Fin |
| 64 | 62, 63 | eqeltri 2837 |
. . . . . . . 8
⊢ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ) ∈
Fin |
| 65 | | unfi 9211 |
. . . . . . . 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 | eqeltrid 2845 |
. . . . . 6
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → (◡(𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) “ ℕ) ∈
Fin) |
| 68 | | cnvimass 6100 |
. . . . . . . . 9
⊢ (◡𝑜 “ ℕ) ⊆ dom 𝑜 |
| 69 | 68, 2 | fssdm 6755 |
. . . . . . . 8
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → (◡𝑜 “ ℕ) ⊆ 𝐽) |
| 70 | | 0ss 4400 |
. . . . . . . . . 10
⊢ ∅
⊆ 𝐽 |
| 71 | 62, 70 | eqsstri 4030 |
. . . . . . . . 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 4192 |
. . . . . . 7
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → ((◡𝑜 “ ℕ) ∪ (◡((ℕ ∖ 𝐽) × {0}) “ ℕ)) ⊆
𝐽) |
| 74 | 29, 73 | eqsstrid 4022 |
. . . . . 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 34371 |
. . . . . 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 1344 |
. . . . 5
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇 ∩ 𝑅)) |
| 84 | | resundir 6012 |
. . . . . 6
⊢ ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽) = ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) |
| 85 | | ffn 6736 |
. . . . . . . 8
⊢ (𝑜:𝐽⟶ℕ0 → 𝑜 Fn 𝐽) |
| 86 | | fnresdm 6687 |
. . . . . . . . 9
⊢ (𝑜 Fn 𝐽 → (𝑜 ↾ 𝐽) = 𝑜) |
| 87 | | disjdifr 4473 |
. . . . . . . . . . 11
⊢ ((ℕ
∖ 𝐽) ∩ 𝐽) = ∅ |
| 88 | | fnconstg 6796 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℕ0 → ((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽)) |
| 89 | | fnresdisj 6688 |
. . . . . . . . . . . 12
⊢
(((ℕ ∖ 𝐽) × {0}) Fn (ℕ ∖ 𝐽) → (((ℕ ∖
𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅)) |
| 90 | 15, 88, 89 | mp2b 10 |
. . . . . . . . . . 11
⊢
(((ℕ ∖ 𝐽) ∩ 𝐽) = ∅ ↔ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅) |
| 91 | 87, 90 | mpbi 230 |
. . . . . . . . . 10
⊢
(((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅ |
| 92 | 91 | a1i 11 |
. . . . . . . . 9
⊢ (𝑜 Fn 𝐽 → (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽) = ∅) |
| 93 | 86, 92 | uneq12d 4169 |
. . . . . . . 8
⊢ (𝑜 Fn 𝐽 → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅)) |
| 94 | 2, 85, 93 | 3syl 18 |
. . . . . . 7
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = (𝑜 ∪ ∅)) |
| 95 | | un0 4394 |
. . . . . . 7
⊢ (𝑜 ∪ ∅) = 𝑜 |
| 96 | 94, 95 | eqtrdi 2793 |
. . . . . 6
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → ((𝑜 ↾ 𝐽) ∪ (((ℕ ∖ 𝐽) × {0}) ↾ 𝐽)) = 𝑜) |
| 97 | 84, 96 | eqtr2id 2790 |
. . . . 5
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽)) |
| 98 | | reseq1 5991 |
. . . . . 6
⊢ (𝑚 = (𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) → (𝑚 ↾ 𝐽) = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽)) |
| 99 | 98 | rspceeqv 3645 |
. . . . 5
⊢ (((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = ((𝑜 ∪ ((ℕ ∖ 𝐽) × {0})) ↾ 𝐽)) → ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)) |
| 100 | 83, 97, 99 | syl2anc 584 |
. . . 4
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) → ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)) |
| 101 | | simpr 484 |
. . . . . . 7
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 = (𝑚 ↾ 𝐽)) |
| 102 | | simpl 482 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑚 ∈ (𝑇 ∩ 𝑅)) |
| 103 | 75, 76, 77, 10, 78, 79, 80, 34, 81 | eulerpartlemt0 34371 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ (𝑇 ∩ 𝑅) ↔ (𝑚 ∈ (ℕ0
↑m ℕ) ∧ (◡𝑚 “ ℕ) ∈ Fin ∧ (◡𝑚 “ ℕ) ⊆ 𝐽)) |
| 104 | 102, 103 | sylib 218 |
. . . . . . . . . . 11
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ∈ (ℕ0
↑m ℕ) ∧ (◡𝑚 “ ℕ) ∈ Fin ∧ (◡𝑚 “ ℕ) ⊆ 𝐽)) |
| 105 | 104 | simp1d 1143 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑚 ∈ (ℕ0
↑m ℕ)) |
| 106 | 22, 23 | elmap 8911 |
. . . . . . . . . 10
⊢ (𝑚 ∈ (ℕ0
↑m ℕ) ↔ 𝑚:ℕ⟶ℕ0) |
| 107 | 105, 106 | sylib 218 |
. . . . . . . . 9
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑚:ℕ⟶ℕ0) |
| 108 | | fssres 6774 |
. . . . . . . . 9
⊢ ((𝑚:ℕ⟶ℕ0 ∧
𝐽 ⊆ ℕ) →
(𝑚 ↾ 𝐽):𝐽⟶ℕ0) |
| 109 | 107, 12, 108 | sylancl 586 |
. . . . . . . 8
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽):𝐽⟶ℕ0) |
| 110 | 10, 23 | rabex2 5341 |
. . . . . . . . 9
⊢ 𝐽 ∈ V |
| 111 | 22, 110 | elmap 8911 |
. . . . . . . 8
⊢ ((𝑚 ↾ 𝐽) ∈ (ℕ0
↑m 𝐽)
↔ (𝑚 ↾ 𝐽):𝐽⟶ℕ0) |
| 112 | 109, 111 | sylibr 234 |
. . . . . . 7
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽) ∈ (ℕ0
↑m 𝐽)) |
| 113 | 101, 112 | eqeltrd 2841 |
. . . . . 6
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 ∈ (ℕ0
↑m 𝐽)) |
| 114 | | ffun 6739 |
. . . . . . . . . 10
⊢ (𝑚:ℕ⟶ℕ0 →
Fun 𝑚) |
| 115 | | respreima 7086 |
. . . . . . . . . 10
⊢ (Fun
𝑚 → (◡(𝑚 ↾ 𝐽) “ ℕ) = ((◡𝑚 “ ℕ) ∩ 𝐽)) |
| 116 | 107, 114,
115 | 3syl 18 |
. . . . . . . . 9
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡(𝑚 ↾ 𝐽) “ ℕ) = ((◡𝑚 “ ℕ) ∩ 𝐽)) |
| 117 | 104 | simp2d 1144 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡𝑚 “ ℕ) ∈
Fin) |
| 118 | | infi 9302 |
. . . . . . . . . 10
⊢ ((◡𝑚 “ ℕ) ∈ Fin → ((◡𝑚 “ ℕ) ∩ 𝐽) ∈ Fin) |
| 119 | 117, 118 | syl 17 |
. . . . . . . . 9
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → ((◡𝑚 “ ℕ) ∩ 𝐽) ∈ Fin) |
| 120 | 116, 119 | eqeltrd 2841 |
. . . . . . . 8
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (◡(𝑚 ↾ 𝐽) “ ℕ) ∈
Fin) |
| 121 | | vex 3484 |
. . . . . . . . . 10
⊢ 𝑚 ∈ V |
| 122 | 121 | resex 6047 |
. . . . . . . . 9
⊢ (𝑚 ↾ 𝐽) ∈ V |
| 123 | | cnveq 5884 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑚 ↾ 𝐽) → ◡𝑓 = ◡(𝑚 ↾ 𝐽)) |
| 124 | 123 | imaeq1d 6077 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑚 ↾ 𝐽) → (◡𝑓 “ ℕ) = (◡(𝑚 ↾ 𝐽) “ ℕ)) |
| 125 | 124 | eleq1d 2826 |
. . . . . . . . 9
⊢ (𝑓 = (𝑚 ↾ 𝐽) → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡(𝑚 ↾ 𝐽) “ ℕ) ∈
Fin)) |
| 126 | 122, 125,
34 | elab2 3682 |
. . . . . . . 8
⊢ ((𝑚 ↾ 𝐽) ∈ 𝑅 ↔ (◡(𝑚 ↾ 𝐽) “ ℕ) ∈
Fin) |
| 127 | 120, 126 | sylibr 234 |
. . . . . . 7
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑚 ↾ 𝐽) ∈ 𝑅) |
| 128 | 101, 127 | eqeltrd 2841 |
. . . . . 6
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → 𝑜 ∈ 𝑅) |
| 129 | 113, 128 | jca 511 |
. . . . 5
⊢ ((𝑚 ∈ (𝑇 ∩ 𝑅) ∧ 𝑜 = (𝑚 ↾ 𝐽)) → (𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅)) |
| 130 | 129 | rexlimiva 3147 |
. . . 4
⊢
(∃𝑚 ∈
(𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽) → (𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅)) |
| 131 | 100, 130 | impbii 209 |
. . 3
⊢ ((𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅) ↔ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)) |
| 132 | 131 | abbii 2809 |
. 2
⊢ {𝑜 ∣ (𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅)} = {𝑜 ∣ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)} |
| 133 | | df-in 3958 |
. 2
⊢
((ℕ0 ↑m 𝐽) ∩ 𝑅) = {𝑜 ∣ (𝑜 ∈ (ℕ0
↑m 𝐽) ∧
𝑜 ∈ 𝑅)} |
| 134 | | eqid 2737 |
. . 3
⊢ (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) = (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) |
| 135 | 134 | rnmpt 5968 |
. 2
⊢ ran
(𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) = {𝑜 ∣ ∃𝑚 ∈ (𝑇 ∩ 𝑅)𝑜 = (𝑚 ↾ 𝐽)} |
| 136 | 132, 133,
135 | 3eqtr4i 2775 |
1
⊢
((ℕ0 ↑m 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) |