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Theorem ackbijnn 15392
Description: Translate the Ackermann bijection ackbij1 9852 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 13945 . . . 4 (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
21hashgf1o 13544 . . 3 (♯ ↾ ω):ω–1-1-onto→ℕ0
3 sneq 4551 . . . . . . . . . 10 (𝑤 = 𝑦 → {𝑤} = {𝑦})
4 pweq 4529 . . . . . . . . . 10 (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
53, 4xpeq12d 5582 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
65cbviunv 4949 . . . . . . . 8 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑧 ({𝑦} × 𝒫 𝑦)
7 iuneq1 4920 . . . . . . . 8 (𝑧 = 𝑥 𝑦𝑧 ({𝑦} × 𝒫 𝑦) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
86, 7syl5eq 2790 . . . . . . 7 (𝑧 = 𝑥 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
98fveq2d 6721 . . . . . 6 (𝑧 = 𝑥 → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
109cbvmptv 5158 . . . . 5 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
1110ackbij1 9852 . . . 4 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12 f1ocnv 6673 . . . . . 6 ((♯ ↾ ω):ω–1-1-onto→ℕ0(♯ ↾ ω):ℕ01-1-onto→ω)
132, 12ax-mp 5 . . . . 5 (♯ ↾ ω):ℕ01-1-onto→ω
14 f1opwfi 8980 . . . . 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 6683 . . . 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 6683 . . 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 4144 . . . . . . . . . 10 (𝒫 ω ∩ Fin) ⊆ Fin
21 f1of 6661 . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . 13 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥))
2423fmpt 6927 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin) ↔ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)⟶(𝒫 ω ∩ Fin))
2522, 24mpbir 234 . . . . . . . . . . 11 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin)
2625rspec 3129 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
2720, 26sselid 3898 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) “ 𝑥) ∈ Fin)
28 snfi 8721 . . . . . . . . . . 11 {𝑤} ∈ Fin
29 cnvimass 5949 . . . . . . . . . . . . . . 15 ((♯ ↾ ω) “ 𝑥) ⊆ dom (♯ ↾ ω)
30 dmhashres 13907 . . . . . . . . . . . . . . 15 dom (♯ ↾ ω) = ω
3129, 30sseqtri 3937 . . . . . . . . . . . . . 14 ((♯ ↾ ω) “ 𝑥) ⊆ ω
32 onfin2 8871 . . . . . . . . . . . . . . 15 ω = (On ∩ Fin)
33 inss2 4144 . . . . . . . . . . . . . . 15 (On ∩ Fin) ⊆ Fin
3432, 33eqsstri 3935 . . . . . . . . . . . . . 14 ω ⊆ Fin
3531, 34sstri 3910 . . . . . . . . . . . . 13 ((♯ ↾ ω) “ 𝑥) ⊆ Fin
36 simpr 488 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥))
3735, 36sselid 3898 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38 pwfi 8856 . . . . . . . . . . . 12 (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
3937, 38sylib 221 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40 xpfi 8942 . . . . . . . . . . 11 (({𝑤} ∈ Fin ∧ 𝒫 𝑤 ∈ Fin) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
4128, 39, 40sylancr 590 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
4241ralrimiva 3105 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43 iunfi 8964 . . . . . . . . 9 ((((♯ ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
4427, 42, 43syl2anc 587 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
45 ficardom 9577 . . . . . . . 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 6737 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
48 hashcard 13922 . . . . . . 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 7793 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) ∈ {𝑤})
51 elsni 4558 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑤} → (1st𝑧) = 𝑤)
5250, 51syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) = 𝑤)
5352rgen 3071 . . . . . . . . . 10 𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
5453rgenw 3073 . . . . . . . . 9 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
55 invdisj 5037 . . . . . . . . 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 15386 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ ((♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)))
58 sneq 4551 . . . . . . . . . 10 (𝑤 = ((♯ ↾ ω)‘𝑦) → {𝑤} = {((♯ ↾ ω)‘𝑦)})
59 pweq 4529 . . . . . . . . . 10 (𝑤 = ((♯ ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 ((♯ ↾ ω)‘𝑦))
6058, 59xpeq12d 5582 . . . . . . . . 9 (𝑤 = ((♯ ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦)))
6160fveq2d 6721 . . . . . . . 8 (𝑤 = ((♯ ↾ ω)‘𝑦) → (♯‘({𝑤} × 𝒫 𝑤)) = (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))))
62 elinel2 4110 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
63 f1of1 6660 . . . . . . . . . 10 ((♯ ↾ ω):ℕ01-1-onto→ω → (♯ ↾ ω):ℕ01-1→ω)
6413, 63ax-mp 5 . . . . . . . . 9 (♯ ↾ ω):ℕ01-1→ω
65 elinel1 4109 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
6665elpwid 4524 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
67 f1ores 6675 . . . . . . . . 9 (((♯ ↾ ω):ℕ01-1→ω ∧ 𝑥 ⊆ ℕ0) → ((♯ ↾ ω) ↾ 𝑥):𝑥1-1-onto→((♯ ↾ ω) “ 𝑥))
6864, 66, 67sylancr 590 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) ↾ 𝑥):𝑥1-1-onto→((♯ ↾ ω) “ 𝑥))
69 fvres 6736 . . . . . . . . 9 (𝑦𝑥 → (((♯ ↾ ω) ↾ 𝑥)‘𝑦) = ((♯ ↾ ω)‘𝑦))
7069adantl 485 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (((♯ ↾ ω) ↾ 𝑥)‘𝑦) = ((♯ ↾ ω)‘𝑦))
71 hashcl 13923 . . . . . . . . 9 (({𝑤} × 𝒫 𝑤) ∈ Fin → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
72 nn0cn 12100 . . . . . . . . 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 15287 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ ((♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))))
75 snfi 8721 . . . . . . . . . 10 {((♯ ↾ ω)‘𝑦)} ∈ Fin
7666sselda 3901 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝑦 ∈ ℕ0)
77 f1of 6661 . . . . . . . . . . . . . . 15 ((♯ ↾ ω):ℕ01-1-onto→ω → (♯ ↾ ω):ℕ0⟶ω)
7813, 77ax-mp 5 . . . . . . . . . . . . . 14 (♯ ↾ ω):ℕ0⟶ω
7978ffvelrni 6903 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ0 → ((♯ ↾ ω)‘𝑦) ∈ ω)
8076, 79syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘𝑦) ∈ ω)
8134, 80sselid 3898 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘𝑦) ∈ Fin)
82 pwfi 8856 . . . . . . . . . . 11 (((♯ ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin)
8381, 82sylib 221 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin)
84 hashxp 14001 . . . . . . . . . 10 (({((♯ ↾ ω)‘𝑦)} ∈ Fin ∧ 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))))
8575, 83, 84sylancr 590 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))))
86 hashsng 13936 . . . . . . . . . . 11 (((♯ ↾ ω)‘𝑦) ∈ ω → (♯‘{((♯ ↾ ω)‘𝑦)}) = 1)
8780, 86syl 17 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘{((♯ ↾ ω)‘𝑦)}) = 1)
88 hashpw 14003 . . . . . . . . . . . 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 6737 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = (♯‘((♯ ↾ ω)‘𝑦)))
91 f1ocnvfv2 7088 . . . . . . . . . . . . . 14 (((♯ ↾ ω):ω–1-1-onto→ℕ0𝑦 ∈ ℕ0) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = 𝑦)
922, 76, 91sylancr 590 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = 𝑦)
9390, 92eqtr3d 2779 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘((♯ ↾ ω)‘𝑦)) = 𝑦)
9493oveq2d 7229 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑(♯‘((♯ ↾ ω)‘𝑦))) = (2↑𝑦))
9589, 94eqtrd 2777 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘𝒫 ((♯ ↾ ω)‘𝑦)) = (2↑𝑦))
9687, 95oveq12d 7231 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
97 2cn 11905 . . . . . . . . . . 11 2 ∈ ℂ
98 expcl 13653 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
9997, 76, 98sylancr 590 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑𝑦) ∈ ℂ)
10099mulid2d 10851 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
10185, 96, 1003eqtrd 2781 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = (2↑𝑦))
102101sumeq2dv 15267 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦𝑥 (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = Σ𝑦𝑥 (2↑𝑦))
10357, 74, 1023eqtrd 2781 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (2↑𝑦))
10447, 49, 1033eqtrd 2781 . . . . 5 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦𝑥 (2↑𝑦))
105104mpteq2ia 5146 . . . 4 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦𝑥 (2↑𝑦))
10646adantl 485 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
10726adantl 485 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → ((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
108 eqidd 2738 . . . . . . 7 (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))
109 eqidd 2738 . . . . . . 7 (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))))
110 iuneq1 4920 . . . . . . . 8 (𝑧 = ((♯ ↾ ω) “ 𝑥) → 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
111110fveq2d 6721 . . . . . . 7 (𝑧 = ((♯ ↾ ω) “ 𝑥) → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
112107, 108, 109, 111fmptco 6944 . . . . . 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 6661 . . . . . . . 8 ((♯ ↾ ω):ω–1-1-onto→ℕ0 → (♯ ↾ ω):ω⟶ℕ0)
1142, 113mp1i 13 . . . . . . 7 (⊤ → (♯ ↾ ω):ω⟶ℕ0)
115114feqmptd 6780 . . . . . 6 (⊤ → (♯ ↾ ω) = (𝑦 ∈ ω ↦ ((♯ ↾ ω)‘𝑦)))
116 fveq2 6717 . . . . . 6 (𝑦 = (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) → ((♯ ↾ ω)‘𝑦) = ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
117106, 112, 115, 116fmptco 6944 . . . . 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 1550 . . . 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 2775 . . 3 ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))) = 𝐹
121 f1oeq1 6649 . . 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 233 1 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wtru 1544  wcel 2110  wral 3061  cin 3865  wss 3866  𝒫 cpw 4513  {csn 4541   ciun 4904  Disj wdisj 5018  cmpt 5135   × cxp 5549  ccnv 5550  dom cdm 5551  cres 5553  cima 5554  ccom 5555  Oncon0 6213  wf 6376  1-1wf1 6377  1-1-ontowf1o 6379  cfv 6380  (class class class)co 7213  ωcom 7644  1st c1st 7759  Fincfn 8626  cardccrd 9551  cc 10727  1c1 10730   · cmul 10734  2c2 11885  0cn0 12090  cexp 13635  chash 13896  Σcsu 15249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-disj 5019  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-oadd 8206  df-er 8391  df-map 8510  df-pm 8511  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-sup 9058  df-oi 9126  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-rp 12587  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-sum 15250
This theorem is referenced by:  bitsinv2  16002
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