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Theorem ackbijnn 14844
Description: Translate the Ackermann bijection ackbij1 9313 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 13369 . . . 4 (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
21hashgf1o 12978 . . 3 (♯ ↾ ω):ω–1-1-onto→ℕ0
3 sneq 4344 . . . . . . . . . 10 (𝑤 = 𝑦 → {𝑤} = {𝑦})
4 pweq 4318 . . . . . . . . . 10 (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
53, 4xpeq12d 5308 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
65cbviunv 4715 . . . . . . . 8 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑧 ({𝑦} × 𝒫 𝑦)
7 iuneq1 4690 . . . . . . . 8 (𝑧 = 𝑥 𝑦𝑧 ({𝑦} × 𝒫 𝑦) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
86, 7syl5eq 2811 . . . . . . 7 (𝑧 = 𝑥 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
98fveq2d 6379 . . . . . 6 (𝑧 = 𝑥 → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
109cbvmptv 4909 . . . . 5 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
1110ackbij1 9313 . . . 4 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12 f1ocnv 6332 . . . . . 6 ((♯ ↾ ω):ω–1-1-onto→ℕ0(♯ ↾ ω):ℕ01-1-onto→ω)
132, 12ax-mp 5 . . . . 5 (♯ ↾ ω):ℕ01-1-onto→ω
14 f1opwfi 8477 . . . . 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 6342 . . . 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 683 . . 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 6342 . . 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 683 . 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 3993 . . . . . . . . . 10 (𝒫 ω ∩ Fin) ⊆ Fin
21 f1of 6320 . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . 13 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥))
2423fmpt 6570 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin) ↔ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)⟶(𝒫 ω ∩ Fin))
2522, 24mpbir 222 . . . . . . . . . . 11 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin)
2625rspec 3078 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
2720, 26sseldi 3759 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) “ 𝑥) ∈ Fin)
28 snfi 8245 . . . . . . . . . . 11 {𝑤} ∈ Fin
29 cnvimass 5667 . . . . . . . . . . . . . . 15 ((♯ ↾ ω) “ 𝑥) ⊆ dom (♯ ↾ ω)
30 dmhashres 13333 . . . . . . . . . . . . . . 15 dom (♯ ↾ ω) = ω
3129, 30sseqtri 3797 . . . . . . . . . . . . . 14 ((♯ ↾ ω) “ 𝑥) ⊆ ω
32 onfin2 8359 . . . . . . . . . . . . . . 15 ω = (On ∩ Fin)
33 inss2 3993 . . . . . . . . . . . . . . 15 (On ∩ Fin) ⊆ Fin
3432, 33eqsstri 3795 . . . . . . . . . . . . . 14 ω ⊆ Fin
3531, 34sstri 3770 . . . . . . . . . . . . 13 ((♯ ↾ ω) “ 𝑥) ⊆ Fin
36 simpr 477 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥))
3735, 36sseldi 3759 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38 pwfi 8468 . . . . . . . . . . . 12 (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
3937, 38sylib 209 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40 xpfi 8438 . . . . . . . . . . 11 (({𝑤} ∈ Fin ∧ 𝒫 𝑤 ∈ Fin) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
4128, 39, 40sylancr 581 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
4241ralrimiva 3113 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43 iunfi 8461 . . . . . . . . 9 ((((♯ ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
4427, 42, 43syl2anc 579 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
45 ficardom 9038 . . . . . . . 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)
47 fvres 6394 . . . . . . 7 ((card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω → ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
4846, 47syl 17 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
49 hashcard 13348 . . . . . . 7 ( 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin → (♯‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
5044, 49syl 17 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
51 xp1st 7398 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) ∈ {𝑤})
52 elsni 4351 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑤} → (1st𝑧) = 𝑤)
5351, 52syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) = 𝑤)
5453rgen 3069 . . . . . . . . . 10 𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
5554rgenw 3071 . . . . . . . . 9 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
56 invdisj 4795 . . . . . . . . 9 (∀𝑤 ∈ ((♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤Disj 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
5755, 56mp1i 13 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Disj 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
5827, 41, 57hashiun 14838 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ ((♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)))
59 sneq 4344 . . . . . . . . . 10 (𝑤 = ((♯ ↾ ω)‘𝑦) → {𝑤} = {((♯ ↾ ω)‘𝑦)})
60 pweq 4318 . . . . . . . . . 10 (𝑤 = ((♯ ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 ((♯ ↾ ω)‘𝑦))
6159, 60xpeq12d 5308 . . . . . . . . 9 (𝑤 = ((♯ ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦)))
6261fveq2d 6379 . . . . . . . 8 (𝑤 = ((♯ ↾ ω)‘𝑦) → (♯‘({𝑤} × 𝒫 𝑤)) = (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))))
63 inss2 3993 . . . . . . . . 9 (𝒫 ℕ0 ∩ Fin) ⊆ Fin
6463sseli 3757 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
65 f1of1 6319 . . . . . . . . . 10 ((♯ ↾ ω):ℕ01-1-onto→ω → (♯ ↾ ω):ℕ01-1→ω)
6613, 65ax-mp 5 . . . . . . . . 9 (♯ ↾ ω):ℕ01-1→ω
67 inss1 3992 . . . . . . . . . . 11 (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0
6867sseli 3757 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
6968elpwid 4327 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
70 f1ores 6334 . . . . . . . . 9 (((♯ ↾ ω):ℕ01-1→ω ∧ 𝑥 ⊆ ℕ0) → ((♯ ↾ ω) ↾ 𝑥):𝑥1-1-onto→((♯ ↾ ω) “ 𝑥))
7166, 69, 70sylancr 581 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) ↾ 𝑥):𝑥1-1-onto→((♯ ↾ ω) “ 𝑥))
72 fvres 6394 . . . . . . . . 9 (𝑦𝑥 → (((♯ ↾ ω) ↾ 𝑥)‘𝑦) = ((♯ ↾ ω)‘𝑦))
7372adantl 473 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (((♯ ↾ ω) ↾ 𝑥)‘𝑦) = ((♯ ↾ ω)‘𝑦))
74 hashcl 13349 . . . . . . . . 9 (({𝑤} × 𝒫 𝑤) ∈ Fin → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
75 nn0cn 11549 . . . . . . . . 9 ((♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0 → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℂ)
7641, 74, 753syl 18 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℂ)
7762, 64, 71, 73, 76fsumf1o 14739 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ ((♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))))
78 snfi 8245 . . . . . . . . . 10 {((♯ ↾ ω)‘𝑦)} ∈ Fin
7969sselda 3761 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝑦 ∈ ℕ0)
80 f1of 6320 . . . . . . . . . . . . . . 15 ((♯ ↾ ω):ℕ01-1-onto→ω → (♯ ↾ ω):ℕ0⟶ω)
8113, 80ax-mp 5 . . . . . . . . . . . . . 14 (♯ ↾ ω):ℕ0⟶ω
8281ffvelrni 6548 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ0 → ((♯ ↾ ω)‘𝑦) ∈ ω)
8379, 82syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘𝑦) ∈ ω)
8434, 83sseldi 3759 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘𝑦) ∈ Fin)
85 pwfi 8468 . . . . . . . . . . 11 (((♯ ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin)
8684, 85sylib 209 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin)
87 hashxp 13422 . . . . . . . . . 10 (({((♯ ↾ ω)‘𝑦)} ∈ Fin ∧ 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))))
8878, 86, 87sylancr 581 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))))
89 hashsng 13361 . . . . . . . . . . 11 (((♯ ↾ ω)‘𝑦) ∈ ω → (♯‘{((♯ ↾ ω)‘𝑦)}) = 1)
9083, 89syl 17 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘{((♯ ↾ ω)‘𝑦)}) = 1)
91 hashpw 13424 . . . . . . . . . . . 12 (((♯ ↾ ω)‘𝑦) ∈ Fin → (♯‘𝒫 ((♯ ↾ ω)‘𝑦)) = (2↑(♯‘((♯ ↾ ω)‘𝑦))))
9284, 91syl 17 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘𝒫 ((♯ ↾ ω)‘𝑦)) = (2↑(♯‘((♯ ↾ ω)‘𝑦))))
93 fvres 6394 . . . . . . . . . . . . . 14 (((♯ ↾ ω)‘𝑦) ∈ ω → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = (♯‘((♯ ↾ ω)‘𝑦)))
9483, 93syl 17 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = (♯‘((♯ ↾ ω)‘𝑦)))
95 f1ocnvfv2 6725 . . . . . . . . . . . . . 14 (((♯ ↾ ω):ω–1-1-onto→ℕ0𝑦 ∈ ℕ0) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = 𝑦)
962, 79, 95sylancr 581 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = 𝑦)
9794, 96eqtr3d 2801 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘((♯ ↾ ω)‘𝑦)) = 𝑦)
9897oveq2d 6858 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑(♯‘((♯ ↾ ω)‘𝑦))) = (2↑𝑦))
9992, 98eqtrd 2799 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘𝒫 ((♯ ↾ ω)‘𝑦)) = (2↑𝑦))
10090, 99oveq12d 6860 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
101 2cn 11347 . . . . . . . . . . 11 2 ∈ ℂ
102 expcl 13085 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
103101, 79, 102sylancr 581 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑𝑦) ∈ ℂ)
104103mulid2d 10312 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
10588, 100, 1043eqtrd 2803 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = (2↑𝑦))
106105sumeq2dv 14718 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦𝑥 (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = Σ𝑦𝑥 (2↑𝑦))
10758, 77, 1063eqtrd 2803 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (2↑𝑦))
10848, 50, 1073eqtrd 2803 . . . . 5 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦𝑥 (2↑𝑦))
109108mpteq2ia 4899 . . . 4 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦𝑥 (2↑𝑦))
11046adantl 473 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
11126adantl 473 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → ((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
112 eqidd 2766 . . . . . . 7 (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))
113 eqidd 2766 . . . . . . 7 (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))))
114 iuneq1 4690 . . . . . . . 8 (𝑧 = ((♯ ↾ ω) “ 𝑥) → 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
115114fveq2d 6379 . . . . . . 7 (𝑧 = ((♯ ↾ ω) “ 𝑥) → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
116111, 112, 113, 115fmptco 6587 . . . . . 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))))
117 f1of 6320 . . . . . . . 8 ((♯ ↾ ω):ω–1-1-onto→ℕ0 → (♯ ↾ ω):ω⟶ℕ0)
1182, 117mp1i 13 . . . . . . 7 (⊤ → (♯ ↾ ω):ω⟶ℕ0)
119118feqmptd 6438 . . . . . 6 (⊤ → (♯ ↾ ω) = (𝑦 ∈ ω ↦ ((♯ ↾ ω)‘𝑦)))
120 fveq2 6375 . . . . . 6 (𝑦 = (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) → ((♯ ↾ ω)‘𝑦) = ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
121110, 116, 119, 120fmptco 6587 . . . . 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)))))
122121mptru 1660 . . . 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))))
123 ackbijnn.1 . . . 4 𝐹 = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦𝑥 (2↑𝑦))
124109, 122, 1233eqtr4i 2797 . . 3 ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))) = 𝐹
125 f1oeq1 6310 . . 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))
126124, 125ax-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)
12719, 126mpbi 221 1 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wtru 1653  wcel 2155  wral 3055  cin 3731  wss 3732  𝒫 cpw 4315  {csn 4334   ciun 4676  Disj wdisj 4777  cmpt 4888   × cxp 5275  ccnv 5276  dom cdm 5277  cres 5279  cima 5280  ccom 5281  Oncon0 5908  wf 6064  1-1wf1 6065  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  ωcom 7263  1st c1st 7364  Fincfn 8160  cardccrd 9012  cc 10187  1c1 10190   · cmul 10194  2c2 11327  0cn0 11538  cexp 13067  chash 13321  Σcsu 14701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-disj 4778  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-xnn0 11611  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-sum 14702
This theorem is referenced by:  bitsinv2  15446
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