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Theorem ackbijnn 15860
Description: Translate the Ackermann bijection ackbij1 10274 onto the positive integers. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypothesis
Ref Expression
ackbijnn.1 𝐹 = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦𝑥 (2↑𝑦))
Assertion
Ref Expression
ackbijnn 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem ackbijnn
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hashgval2 14413 . . . 4 (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
21hashgf1o 14008 . . 3 (♯ ↾ ω):ω–1-1-onto→ℕ0
3 sneq 4640 . . . . . . . . . 10 (𝑤 = 𝑦 → {𝑤} = {𝑦})
4 pweq 4618 . . . . . . . . . 10 (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
53, 4xpeq12d 5719 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
65cbviunv 5044 . . . . . . . 8 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑧 ({𝑦} × 𝒫 𝑦)
7 iuneq1 5012 . . . . . . . 8 (𝑧 = 𝑥 𝑦𝑧 ({𝑦} × 𝒫 𝑦) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
86, 7eqtrid 2786 . . . . . . 7 (𝑧 = 𝑥 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
98fveq2d 6910 . . . . . 6 (𝑧 = 𝑥 → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
109cbvmptv 5260 . . . . 5 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
1110ackbij1 10274 . . . 4 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12 f1ocnv 6860 . . . . . 6 ((♯ ↾ ω):ω–1-1-onto→ℕ0(♯ ↾ ω):ℕ01-1-onto→ω)
132, 12ax-mp 5 . . . . 5 (♯ ↾ ω):ℕ01-1-onto→ω
14 f1opwfi 9393 . . . . 5 ((♯ ↾ ω):ℕ01-1-onto→ω → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)–1-1-onto→(𝒫 ω ∩ Fin))
1513, 14ax-mp 5 . . . 4 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)–1-1-onto→(𝒫 ω ∩ Fin)
16 f1oco 6871 . . . 4 (((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({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‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ω)
1711, 15, 16mp2an 692 . . 3 ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ω
18 f1oco 6871 . . 3 (((♯ ↾ ω):ω–1-1-onto→ℕ0 ∧ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ω) → ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0)
192, 17, 18mp2an 692 . 2 ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0
20 inss2 4245 . . . . . . . . . 10 (𝒫 ω ∩ Fin) ⊆ Fin
21 f1of 6848 . . . . . . . . . . . . 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))
2215, 21ax-mp 5 . . . . . . . . . . . 12 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)⟶(𝒫 ω ∩ Fin)
23 eqid 2734 . . . . . . . . . . . . 13 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥))
2423fmpt 7129 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin) ↔ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)⟶(𝒫 ω ∩ Fin))
2522, 24mpbir 231 . . . . . . . . . . 11 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin)
2625rspec 3247 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
2720, 26sselid 3992 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) “ 𝑥) ∈ Fin)
28 snfi 9081 . . . . . . . . . . 11 {𝑤} ∈ Fin
29 cnvimass 6101 . . . . . . . . . . . . . . 15 ((♯ ↾ ω) “ 𝑥) ⊆ dom (♯ ↾ ω)
30 dmhashres 14376 . . . . . . . . . . . . . . 15 dom (♯ ↾ ω) = ω
3129, 30sseqtri 4031 . . . . . . . . . . . . . 14 ((♯ ↾ ω) “ 𝑥) ⊆ ω
32 onfin2 9265 . . . . . . . . . . . . . . 15 ω = (On ∩ Fin)
33 inss2 4245 . . . . . . . . . . . . . . 15 (On ∩ Fin) ⊆ Fin
3432, 33eqsstri 4029 . . . . . . . . . . . . . 14 ω ⊆ Fin
3531, 34sstri 4004 . . . . . . . . . . . . 13 ((♯ ↾ ω) “ 𝑥) ⊆ Fin
36 simpr 484 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥))
3735, 36sselid 3992 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38 pwfi 9354 . . . . . . . . . . . 12 (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
3937, 38sylib 218 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40 xpfi 9355 . . . . . . . . . . 11 (({𝑤} ∈ Fin ∧ 𝒫 𝑤 ∈ Fin) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
4128, 39, 40sylancr 587 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
4241ralrimiva 3143 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43 iunfi 9380 . . . . . . . . 9 ((((♯ ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
4427, 42, 43syl2anc 584 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
45 ficardom 9998 . . . . . . . 8 ( 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin → (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
4644, 45syl 17 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
4746fvresd 6926 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
48 hashcard 14390 . . . . . . 7 ( 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin → (♯‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
4944, 48syl 17 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
50 xp1st 8044 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) ∈ {𝑤})
51 elsni 4647 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑤} → (1st𝑧) = 𝑤)
5250, 51syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) = 𝑤)
5352rgen 3060 . . . . . . . . . 10 𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
5453rgenw 3062 . . . . . . . . 9 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
55 invdisj 5133 . . . . . . . . 9 (∀𝑤 ∈ ((♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤Disj 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
5654, 55mp1i 13 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Disj 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
5727, 41, 56hashiun 15854 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ ((♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)))
58 sneq 4640 . . . . . . . . . 10 (𝑤 = ((♯ ↾ ω)‘𝑦) → {𝑤} = {((♯ ↾ ω)‘𝑦)})
59 pweq 4618 . . . . . . . . . 10 (𝑤 = ((♯ ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 ((♯ ↾ ω)‘𝑦))
6058, 59xpeq12d 5719 . . . . . . . . 9 (𝑤 = ((♯ ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦)))
6160fveq2d 6910 . . . . . . . 8 (𝑤 = ((♯ ↾ ω)‘𝑦) → (♯‘({𝑤} × 𝒫 𝑤)) = (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))))
62 elinel2 4211 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
63 f1of1 6847 . . . . . . . . . 10 ((♯ ↾ ω):ℕ01-1-onto→ω → (♯ ↾ ω):ℕ01-1→ω)
6413, 63ax-mp 5 . . . . . . . . 9 (♯ ↾ ω):ℕ01-1→ω
65 elinel1 4210 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
6665elpwid 4613 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
67 f1ores 6862 . . . . . . . . 9 (((♯ ↾ ω):ℕ01-1→ω ∧ 𝑥 ⊆ ℕ0) → ((♯ ↾ ω) ↾ 𝑥):𝑥1-1-onto→((♯ ↾ ω) “ 𝑥))
6864, 66, 67sylancr 587 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) ↾ 𝑥):𝑥1-1-onto→((♯ ↾ ω) “ 𝑥))
69 fvres 6925 . . . . . . . . 9 (𝑦𝑥 → (((♯ ↾ ω) ↾ 𝑥)‘𝑦) = ((♯ ↾ ω)‘𝑦))
7069adantl 481 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (((♯ ↾ ω) ↾ 𝑥)‘𝑦) = ((♯ ↾ ω)‘𝑦))
71 hashcl 14391 . . . . . . . . 9 (({𝑤} × 𝒫 𝑤) ∈ Fin → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
72 nn0cn 12533 . . . . . . . . 9 ((♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0 → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℂ)
7341, 71, 723syl 18 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℂ)
7461, 62, 68, 70, 73fsumf1o 15755 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ ((♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))))
75 snfi 9081 . . . . . . . . . 10 {((♯ ↾ ω)‘𝑦)} ∈ Fin
7666sselda 3994 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝑦 ∈ ℕ0)
77 f1of 6848 . . . . . . . . . . . . . . 15 ((♯ ↾ ω):ℕ01-1-onto→ω → (♯ ↾ ω):ℕ0⟶ω)
7813, 77ax-mp 5 . . . . . . . . . . . . . 14 (♯ ↾ ω):ℕ0⟶ω
7978ffvelcdmi 7102 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ0 → ((♯ ↾ ω)‘𝑦) ∈ ω)
8076, 79syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘𝑦) ∈ ω)
8134, 80sselid 3992 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘𝑦) ∈ Fin)
82 pwfi 9354 . . . . . . . . . . 11 (((♯ ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin)
8381, 82sylib 218 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin)
84 hashxp 14469 . . . . . . . . . 10 (({((♯ ↾ ω)‘𝑦)} ∈ Fin ∧ 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))))
8575, 83, 84sylancr 587 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))))
86 hashsng 14404 . . . . . . . . . . 11 (((♯ ↾ ω)‘𝑦) ∈ ω → (♯‘{((♯ ↾ ω)‘𝑦)}) = 1)
8780, 86syl 17 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘{((♯ ↾ ω)‘𝑦)}) = 1)
88 hashpw 14471 . . . . . . . . . . . 12 (((♯ ↾ ω)‘𝑦) ∈ Fin → (♯‘𝒫 ((♯ ↾ ω)‘𝑦)) = (2↑(♯‘((♯ ↾ ω)‘𝑦))))
8981, 88syl 17 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘𝒫 ((♯ ↾ ω)‘𝑦)) = (2↑(♯‘((♯ ↾ ω)‘𝑦))))
9080fvresd 6926 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = (♯‘((♯ ↾ ω)‘𝑦)))
91 f1ocnvfv2 7296 . . . . . . . . . . . . . 14 (((♯ ↾ ω):ω–1-1-onto→ℕ0𝑦 ∈ ℕ0) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = 𝑦)
922, 76, 91sylancr 587 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = 𝑦)
9390, 92eqtr3d 2776 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘((♯ ↾ ω)‘𝑦)) = 𝑦)
9493oveq2d 7446 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑(♯‘((♯ ↾ ω)‘𝑦))) = (2↑𝑦))
9589, 94eqtrd 2774 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘𝒫 ((♯ ↾ ω)‘𝑦)) = (2↑𝑦))
9687, 95oveq12d 7448 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
97 2cn 12338 . . . . . . . . . . 11 2 ∈ ℂ
98 expcl 14116 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
9997, 76, 98sylancr 587 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑𝑦) ∈ ℂ)
10099mullidd 11276 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
10185, 96, 1003eqtrd 2778 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = (2↑𝑦))
102101sumeq2dv 15734 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦𝑥 (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = Σ𝑦𝑥 (2↑𝑦))
10357, 74, 1023eqtrd 2778 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (2↑𝑦))
10447, 49, 1033eqtrd 2778 . . . . 5 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦𝑥 (2↑𝑦))
105104mpteq2ia 5250 . . . 4 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦𝑥 (2↑𝑦))
10646adantl 481 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
10726adantl 481 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → ((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
108 eqidd 2735 . . . . . . 7 (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))
109 eqidd 2735 . . . . . . 7 (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))))
110 iuneq1 5012 . . . . . . . 8 (𝑧 = ((♯ ↾ ω) “ 𝑥) → 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
111110fveq2d 6910 . . . . . . 7 (𝑧 = ((♯ ↾ ω) “ 𝑥) → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
112107, 108, 109, 111fmptco 7148 . . . . . 6 (⊤ → ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
113 f1of 6848 . . . . . . . 8 ((♯ ↾ ω):ω–1-1-onto→ℕ0 → (♯ ↾ ω):ω⟶ℕ0)
1142, 113mp1i 13 . . . . . . 7 (⊤ → (♯ ↾ ω):ω⟶ℕ0)
115114feqmptd 6976 . . . . . 6 (⊤ → (♯ ↾ ω) = (𝑦 ∈ ω ↦ ((♯ ↾ ω)‘𝑦)))
116 fveq2 6906 . . . . . 6 (𝑦 = (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) → ((♯ ↾ ω)‘𝑦) = ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
117106, 112, 115, 116fmptco 7148 . . . . 5 (⊤ → ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))))
118117mptru 1543 . . . 4 ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
119 ackbijnn.1 . . . 4 𝐹 = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦𝑥 (2↑𝑦))
120105, 118, 1193eqtr4i 2772 . . 3 ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))) = 𝐹
121 f1oeq1 6836 . . 3 (((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))) = 𝐹 → (((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0))
122120, 121ax-mp 5 . 2 (((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0)
12319, 122mpbi 230 1 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wtru 1537  wcel 2105  wral 3058  cin 3961  wss 3962  𝒫 cpw 4604  {csn 4630   ciun 4995  Disj wdisj 5114  cmpt 5230   × cxp 5686  ccnv 5687  dom cdm 5688  cres 5690  cima 5691  ccom 5692  Oncon0 6385  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  ωcom 7886  1st c1st 8010  Fincfn 8983  cardccrd 9972  cc 11150  1c1 11153   · cmul 11157  2c2 12318  0cn0 12523  cexp 14098  chash 14365  Σcsu 15718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719
This theorem is referenced by:  bitsinv2  16476
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