Step | Hyp | Ref
| Expression |
1 | | hashgval2 13028 |
. . . 4
⊢ (#
↾ ω) = (rec((𝑥
∈ V ↦ (𝑥 + 1)),
0) ↾ ω) |
2 | 1 | hashgf1o 12632 |
. . 3
⊢ (#
↾ ω):ω–1-1-onto→ℕ0 |
3 | | sneq 4135 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦}) |
4 | | pweq 4111 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦) |
5 | 3, 4 | xpeq12d 5064 |
. . . . . . . . 9
⊢ (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦)) |
6 | 5 | cbviunv 4495 |
. . . . . . . 8
⊢ ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦) |
7 | | iuneq1 4470 |
. . . . . . . 8
⊢ (𝑧 = 𝑥 → ∪
𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)) |
8 | 6, 7 | syl5eq 2656 |
. . . . . . 7
⊢ (𝑧 = 𝑥 → ∪
𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)) |
9 | 8 | fveq2d 6107 |
. . . . . 6
⊢ (𝑧 = 𝑥 → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) |
10 | 9 | cbvmptv 4678 |
. . . . 5
⊢ (𝑧 ∈ (𝒫 ω ∩
Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦
(card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) |
11 | 10 | ackbij1 8943 |
. . . 4
⊢ (𝑧 ∈ (𝒫 ω ∩
Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω |
12 | | f1ocnv 6062 |
. . . . . 6
⊢ ((#
↾ ω):ω–1-1-onto→ℕ0 → ◡(# ↾
ω):ℕ0–1-1-onto→ω) |
13 | 2, 12 | ax-mp 5 |
. . . . 5
⊢ ◡(# ↾
ω):ℕ0–1-1-onto→ω |
14 | | f1opwfi 8153 |
. . . . 5
⊢ (◡(# ↾
ω):ℕ0–1-1-onto→ω → (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥)):(𝒫
ℕ0 ∩ Fin)–1-1-onto→(𝒫 ω ∩
Fin)) |
15 | 13, 14 | ax-mp 5 |
. . . 4
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)):(𝒫
ℕ0 ∩ Fin)–1-1-onto→(𝒫 ω ∩ Fin) |
16 | | f1oco 6072 |
. . . 4
⊢ (((𝑧 ∈ (𝒫 ω ∩
Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω ∧ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥)):(𝒫
ℕ0 ∩ Fin)–1-1-onto→(𝒫 ω ∩ Fin)) → ((𝑧 ∈ (𝒫 ω ∩
Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥))):(𝒫
ℕ0 ∩ Fin)–1-1-onto→ω) |
17 | 11, 15, 16 | mp2an 704 |
. . 3
⊢ ((𝑧 ∈ (𝒫 ω ∩
Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥))):(𝒫
ℕ0 ∩ Fin)–1-1-onto→ω |
18 | | f1oco 6072 |
. . 3
⊢ (((#
↾ ω):ω–1-1-onto→ℕ0 ∧ ((𝑧 ∈ (𝒫 ω ∩
Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥))):(𝒫
ℕ0 ∩ Fin)–1-1-onto→ω) → ((# ↾ ω) ∘
((𝑧 ∈ (𝒫
ω ∩ Fin) ↦ (card‘∪
𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥)))):(𝒫
ℕ0 ∩ Fin)–1-1-onto→ℕ0) |
19 | 2, 17, 18 | mp2an 704 |
. 2
⊢ ((#
↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦
(card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥)))):(𝒫
ℕ0 ∩ Fin)–1-1-onto→ℕ0 |
20 | | inss2 3796 |
. . . . . . . . . 10
⊢
(𝒫 ω ∩ Fin) ⊆ Fin |
21 | | f1of 6050 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)):(𝒫
ℕ0 ∩ Fin)–1-1-onto→(𝒫 ω ∩ Fin) → (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)):(𝒫
ℕ0 ∩ Fin)⟶(𝒫 ω ∩
Fin)) |
22 | 15, 21 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)):(𝒫
ℕ0 ∩ Fin)⟶(𝒫 ω ∩
Fin) |
23 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥)) |
24 | 23 | fmpt 6289 |
. . . . . . . . . . . 12
⊢
(∀𝑥 ∈
(𝒫 ℕ0 ∩ Fin)(◡(# ↾ ω) “ 𝑥) ∈ (𝒫 ω
∩ Fin) ↔ (𝑥 ∈
(𝒫 ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)):(𝒫
ℕ0 ∩ Fin)⟶(𝒫 ω ∩
Fin)) |
25 | 22, 24 | mpbir 220 |
. . . . . . . . . . 11
⊢
∀𝑥 ∈
(𝒫 ℕ0 ∩ Fin)(◡(# ↾ ω) “ 𝑥) ∈ (𝒫 ω
∩ Fin) |
26 | 25 | rspec 2915 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → (◡(# ↾ ω) “ 𝑥) ∈ (𝒫 ω
∩ Fin)) |
27 | 20, 26 | sseldi 3566 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → (◡(# ↾ ω) “ 𝑥) ∈ Fin) |
28 | | snfi 7923 |
. . . . . . . . . . 11
⊢ {𝑤} ∈ Fin |
29 | | cnvimass 5404 |
. . . . . . . . . . . . . . 15
⊢ (◡(# ↾ ω) “ 𝑥) ⊆ dom (# ↾
ω) |
30 | | dmhashres 12991 |
. . . . . . . . . . . . . . 15
⊢ dom (#
↾ ω) = ω |
31 | 29, 30 | sseqtri 3600 |
. . . . . . . . . . . . . 14
⊢ (◡(# ↾ ω) “ 𝑥) ⊆
ω |
32 | | onfin2 8037 |
. . . . . . . . . . . . . . 15
⊢ ω =
(On ∩ Fin) |
33 | | inss2 3796 |
. . . . . . . . . . . . . . 15
⊢ (On ∩
Fin) ⊆ Fin |
34 | 32, 33 | eqsstri 3598 |
. . . . . . . . . . . . . 14
⊢ ω
⊆ Fin |
35 | 31, 34 | sstri 3577 |
. . . . . . . . . . . . 13
⊢ (◡(# ↾ ω) “ 𝑥) ⊆ Fin |
36 | | simpr 476 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) |
37 | 35, 36 | sseldi 3566 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin) |
38 | | pwfi 8144 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ Fin ↔ 𝒫
𝑤 ∈
Fin) |
39 | 37, 38 | sylib 207 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin) |
40 | | xpfi 8116 |
. . . . . . . . . . 11
⊢ (({𝑤} ∈ Fin ∧ 𝒫
𝑤 ∈ Fin) →
({𝑤} × 𝒫
𝑤) ∈
Fin) |
41 | 28, 39, 40 | sylancr 694 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin) |
42 | 41 | ralrimiva 2949 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → ∀𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) |
43 | | iunfi 8137 |
. . . . . . . . 9
⊢ (((◡(# ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → ∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) |
44 | 27, 42, 43 | syl2anc 691 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → ∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) |
45 | | ficardom 8670 |
. . . . . . . 8
⊢ (∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin → (card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω) |
46 | 44, 45 | syl 17 |
. . . . . . 7
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → (card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω) |
47 | | fvres 6117 |
. . . . . . 7
⊢
((card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω → ((# ↾
ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (#‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) |
48 | 46, 47 | syl 17 |
. . . . . 6
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → ((# ↾
ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (#‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) |
49 | | hashcard 13008 |
. . . . . . 7
⊢ (∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin →
(#‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (#‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) |
50 | 44, 49 | syl 17 |
. . . . . 6
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → (#‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (#‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) |
51 | | xp1st 7089 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) ∈ {𝑤}) |
52 | | elsni 4142 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑧) ∈ {𝑤} → (1st ‘𝑧) = 𝑤) |
53 | 51, 52 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) = 𝑤) |
54 | 53 | rgen 2906 |
. . . . . . . . . 10
⊢
∀𝑧 ∈
({𝑤} × 𝒫
𝑤)(1st
‘𝑧) = 𝑤 |
55 | 54 | rgenw 2908 |
. . . . . . . . 9
⊢
∀𝑤 ∈
(◡(# ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤 |
56 | | invdisj 4571 |
. . . . . . . . 9
⊢
(∀𝑤 ∈
(◡(# ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤 → Disj 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) |
57 | 55, 56 | mp1i 13 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → Disj 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) |
58 | 27, 41, 57 | hashiun 14395 |
. . . . . . 7
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → (#‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ (◡(# ↾ ω) “ 𝑥)(#‘({𝑤} × 𝒫 𝑤))) |
59 | | sneq 4135 |
. . . . . . . . . 10
⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → {𝑤} = {(◡(# ↾ ω)‘𝑦)}) |
60 | | pweq 4111 |
. . . . . . . . . 10
⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 (◡(# ↾ ω)‘𝑦)) |
61 | 59, 60 | xpeq12d 5064 |
. . . . . . . . 9
⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) |
62 | 61 | fveq2d 6107 |
. . . . . . . 8
⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → (#‘({𝑤} × 𝒫 𝑤)) = (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦)))) |
63 | | inss2 3796 |
. . . . . . . . 9
⊢
(𝒫 ℕ0 ∩ Fin) ⊆ Fin |
64 | 63 | sseli 3564 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → 𝑥 ∈ Fin) |
65 | | f1of1 6049 |
. . . . . . . . . 10
⊢ (◡(# ↾
ω):ℕ0–1-1-onto→ω → ◡(# ↾
ω):ℕ0–1-1→ω) |
66 | 13, 65 | ax-mp 5 |
. . . . . . . . 9
⊢ ◡(# ↾
ω):ℕ0–1-1→ω |
67 | | inss1 3795 |
. . . . . . . . . . 11
⊢
(𝒫 ℕ0 ∩ Fin) ⊆ 𝒫
ℕ0 |
68 | 67 | sseli 3564 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫
ℕ0) |
69 | 68 | elpwid 4118 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → 𝑥 ⊆
ℕ0) |
70 | | f1ores 6064 |
. . . . . . . . 9
⊢ ((◡(# ↾
ω):ℕ0–1-1→ω ∧ 𝑥 ⊆ ℕ0) → (◡(# ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(#
↾ ω) “ 𝑥)) |
71 | 66, 69, 70 | sylancr 694 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → (◡(# ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(#
↾ ω) “ 𝑥)) |
72 | | fvres 6117 |
. . . . . . . . 9
⊢ (𝑦 ∈ 𝑥 → ((◡(# ↾ ω) ↾ 𝑥)‘𝑦) = (◡(# ↾ ω)‘𝑦)) |
73 | 72 | adantl 481 |
. . . . . . . 8
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((◡(# ↾ ω) ↾ 𝑥)‘𝑦) = (◡(# ↾ ω)‘𝑦)) |
74 | | hashcl 13009 |
. . . . . . . . 9
⊢ (({𝑤} × 𝒫 𝑤) ∈ Fin →
(#‘({𝑤} ×
𝒫 𝑤)) ∈
ℕ0) |
75 | | nn0cn 11179 |
. . . . . . . . 9
⊢
((#‘({𝑤}
× 𝒫 𝑤))
∈ ℕ0 → (#‘({𝑤} × 𝒫 𝑤)) ∈ ℂ) |
76 | 41, 74, 75 | 3syl 18 |
. . . . . . . 8
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → (#‘({𝑤} × 𝒫 𝑤)) ∈
ℂ) |
77 | 62, 64, 71, 73, 76 | fsumf1o 14301 |
. . . . . . 7
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → Σ𝑤 ∈ (◡(# ↾ ω) “ 𝑥)(#‘({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦)))) |
78 | | snfi 7923 |
. . . . . . . . . 10
⊢ {(◡(# ↾ ω)‘𝑦)} ∈ Fin |
79 | 69 | sselda 3568 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℕ0) |
80 | | f1of 6050 |
. . . . . . . . . . . . . . 15
⊢ (◡(# ↾
ω):ℕ0–1-1-onto→ω → ◡(# ↾
ω):ℕ0⟶ω) |
81 | 13, 80 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ◡(# ↾
ω):ℕ0⟶ω |
82 | 81 | ffvelrni 6266 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ0
→ (◡(# ↾
ω)‘𝑦) ∈
ω) |
83 | 79, 82 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (◡(# ↾ ω)‘𝑦) ∈ ω) |
84 | 34, 83 | sseldi 3566 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (◡(# ↾ ω)‘𝑦) ∈ Fin) |
85 | | pwfi 8144 |
. . . . . . . . . . 11
⊢ ((◡(# ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 (◡(# ↾ ω)‘𝑦) ∈ Fin) |
86 | 84, 85 | sylib 207 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝒫 (◡(# ↾ ω)‘𝑦) ∈ Fin) |
87 | | hashxp 13081 |
. . . . . . . . . 10
⊢ (({(◡(# ↾ ω)‘𝑦)} ∈ Fin ∧ 𝒫 (◡(# ↾ ω)‘𝑦) ∈ Fin) → (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = ((#‘{(◡(# ↾ ω)‘𝑦)}) · (#‘𝒫 (◡(# ↾ ω)‘𝑦)))) |
88 | 78, 86, 87 | sylancr 694 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = ((#‘{(◡(# ↾ ω)‘𝑦)}) · (#‘𝒫 (◡(# ↾ ω)‘𝑦)))) |
89 | | hashsng 13020 |
. . . . . . . . . . 11
⊢ ((◡(# ↾ ω)‘𝑦) ∈ ω → (#‘{(◡(# ↾ ω)‘𝑦)}) = 1) |
90 | 83, 89 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘{(◡(# ↾ ω)‘𝑦)}) = 1) |
91 | | hashpw 13083 |
. . . . . . . . . . . 12
⊢ ((◡(# ↾ ω)‘𝑦) ∈ Fin → (#‘𝒫 (◡(# ↾ ω)‘𝑦)) = (2↑(#‘(◡(# ↾ ω)‘𝑦)))) |
92 | 84, 91 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘𝒫 (◡(# ↾ ω)‘𝑦)) = (2↑(#‘(◡(# ↾ ω)‘𝑦)))) |
93 | | fvres 6117 |
. . . . . . . . . . . . . 14
⊢ ((◡(# ↾ ω)‘𝑦) ∈ ω → ((# ↾
ω)‘(◡(# ↾
ω)‘𝑦)) =
(#‘(◡(# ↾
ω)‘𝑦))) |
94 | 83, 93 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((# ↾ ω)‘(◡(# ↾ ω)‘𝑦)) = (#‘(◡(# ↾ ω)‘𝑦))) |
95 | | f1ocnvfv2 6433 |
. . . . . . . . . . . . . 14
⊢ (((#
↾ ω):ω–1-1-onto→ℕ0 ∧ 𝑦 ∈ ℕ0)
→ ((# ↾ ω)‘(◡(#
↾ ω)‘𝑦))
= 𝑦) |
96 | 2, 79, 95 | sylancr 694 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((# ↾ ω)‘(◡(# ↾ ω)‘𝑦)) = 𝑦) |
97 | 94, 96 | eqtr3d 2646 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘(◡(# ↾ ω)‘𝑦)) = 𝑦) |
98 | 97 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑(#‘(◡(# ↾ ω)‘𝑦))) = (2↑𝑦)) |
99 | 92, 98 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘𝒫 (◡(# ↾ ω)‘𝑦)) = (2↑𝑦)) |
100 | 90, 99 | oveq12d 6567 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((#‘{(◡(# ↾ ω)‘𝑦)}) · (#‘𝒫 (◡(# ↾ ω)‘𝑦))) = (1 · (2↑𝑦))) |
101 | | 2cn 10968 |
. . . . . . . . . . 11
⊢ 2 ∈
ℂ |
102 | | expcl 12740 |
. . . . . . . . . . 11
⊢ ((2
∈ ℂ ∧ 𝑦
∈ ℕ0) → (2↑𝑦) ∈ ℂ) |
103 | 101, 79, 102 | sylancr 694 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑𝑦) ∈ ℂ) |
104 | 103 | mulid2d 9937 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (1 · (2↑𝑦)) = (2↑𝑦)) |
105 | 88, 100, 104 | 3eqtrd 2648 |
. . . . . . . 8
⊢ ((𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = (2↑𝑦)) |
106 | 105 | sumeq2dv 14281 |
. . . . . . 7
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → Σ𝑦 ∈ 𝑥 (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = Σ𝑦 ∈ 𝑥 (2↑𝑦)) |
107 | 58, 77, 106 | 3eqtrd 2648 |
. . . . . 6
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → (#‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (2↑𝑦)) |
108 | 48, 50, 107 | 3eqtrd 2648 |
. . . . 5
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) → ((# ↾
ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦 ∈ 𝑥 (2↑𝑦)) |
109 | 108 | mpteq2ia 4668 |
. . . 4
⊢ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ ((# ↾
ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ Σ𝑦
∈ 𝑥 (2↑𝑦)) |
110 | 46 | adantl 481 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (𝒫 ℕ0 ∩ Fin)) → (card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω) |
111 | 26 | adantl 481 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (𝒫 ℕ0 ∩ Fin)) → (◡(# ↾ ω) “ 𝑥) ∈ (𝒫 ω
∩ Fin)) |
112 | | eqidd 2611 |
. . . . . . 7
⊢ (⊤
→ (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥))) |
113 | | eqidd 2611 |
. . . . . . 7
⊢ (⊤
→ (𝑧 ∈ (𝒫
ω ∩ Fin) ↦ (card‘∪
𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦
(card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))) |
114 | | iuneq1 4470 |
. . . . . . . 8
⊢ (𝑧 = (◡(# ↾ ω) “ 𝑥) → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) |
115 | 114 | fveq2d 6107 |
. . . . . . 7
⊢ (𝑧 = (◡(# ↾ ω) “ 𝑥) → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) |
116 | 111, 112,
113, 115 | fmptco 6303 |
. . . . . 6
⊢ (⊤
→ ((𝑧 ∈
(𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥))) = (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ (card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) |
117 | | f1of 6050 |
. . . . . . . 8
⊢ ((#
↾ ω):ω–1-1-onto→ℕ0 → (# ↾
ω):ω⟶ℕ0) |
118 | 2, 117 | mp1i 13 |
. . . . . . 7
⊢ (⊤
→ (# ↾ ω):ω⟶ℕ0) |
119 | 118 | feqmptd 6159 |
. . . . . 6
⊢ (⊤
→ (# ↾ ω) = (𝑦 ∈ ω ↦ ((# ↾
ω)‘𝑦))) |
120 | | fveq2 6103 |
. . . . . 6
⊢ (𝑦 = (card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) → ((# ↾ ω)‘𝑦) = ((# ↾
ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) |
121 | 110, 116,
119, 120 | fmptco 6303 |
. . . . 5
⊢ (⊤
→ ((# ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦
(card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥)))) = (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ ((# ↾
ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))) |
122 | 121 | trud 1484 |
. . . 4
⊢ ((#
↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦
(card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥)))) = (𝑥 ∈ (𝒫
ℕ0 ∩ Fin) ↦ ((# ↾
ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) |
123 | | ackbijnn.1 |
. . . 4
⊢ 𝐹 = (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ Σ𝑦
∈ 𝑥 (2↑𝑦)) |
124 | 109, 122,
123 | 3eqtr4i 2642 |
. . 3
⊢ ((#
↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦
(card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥)))) = 𝐹 |
125 | | f1oeq1 6040 |
. . 3
⊢ (((#
↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦
(card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥)))) = 𝐹 → (((# ↾ ω)
∘ ((𝑧 ∈
(𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥)))):(𝒫
ℕ0 ∩ Fin)–1-1-onto→ℕ0 ↔ 𝐹:(𝒫 ℕ0 ∩
Fin)–1-1-onto→ℕ0)) |
126 | 124, 125 | ax-mp 5 |
. 2
⊢ (((#
↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦
(card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩
Fin) ↦ (◡(# ↾ ω)
“ 𝑥)))):(𝒫
ℕ0 ∩ Fin)–1-1-onto→ℕ0 ↔ 𝐹:(𝒫 ℕ0 ∩
Fin)–1-1-onto→ℕ0) |
127 | 19, 126 | mpbi 219 |
1
⊢ 𝐹:(𝒫 ℕ0
∩ Fin)–1-1-onto→ℕ0 |