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Theorem ackbijnn 14399
 Description: Translate the Ackermann bijection ackbij1 8943 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 13028 . . . 4 (# ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
21hashgf1o 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)
53, 4xpeq12d 5064 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
65cbviunv 4495 . . . . . . . 8 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑧 ({𝑦} × 𝒫 𝑦)
7 iuneq1 4470 . . . . . . . 8 (𝑧 = 𝑥 𝑦𝑧 ({𝑦} × 𝒫 𝑦) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
86, 7syl5eq 2656 . . . . . . 7 (𝑧 = 𝑥 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
98fveq2d 6107 . . . . . 6 (𝑧 = 𝑥 → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
109cbvmptv 4678 . . . . 5 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
1110ackbij1 8943 . . . 4 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12 f1ocnv 6062 . . . . . 6 ((# ↾ ω):ω–1-1-onto→ℕ0(# ↾ ω):ℕ01-1-onto→ω)
132, 12ax-mp 5 . . . . 5 (# ↾ ω):ℕ01-1-onto→ω
14 f1opwfi 8153 . . . . 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 6072 . . . 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 704 . . 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 6072 . . 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 704 . 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 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))
2215, 21ax-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))
2423fmpt 6289 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin) ↔ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)⟶(𝒫 ω ∩ Fin))
2522, 24mpbir 220 . . . . . . . . . . 11 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin)
2625rspec 2915 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
2720, 26sseldi 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
3129, 30sseqtri 3600 . . . . . . . . . . . . . 14 ((# ↾ ω) “ 𝑥) ⊆ ω
32 onfin2 8037 . . . . . . . . . . . . . . 15 ω = (On ∩ Fin)
33 inss2 3796 . . . . . . . . . . . . . . 15 (On ∩ Fin) ⊆ Fin
3432, 33eqsstri 3598 . . . . . . . . . . . . . 14 ω ⊆ Fin
3531, 34sstri 3577 . . . . . . . . . . . . 13 ((# ↾ ω) “ 𝑥) ⊆ Fin
36 simpr 476 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((# ↾ ω) “ 𝑥)) → 𝑤 ∈ ((# ↾ ω) “ 𝑥))
3735, 36sseldi 3566 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((# ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38 pwfi 8144 . . . . . . . . . . . 12 (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
3937, 38sylib 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)
4128, 39, 40sylancr 694 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((# ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
4241ralrimiva 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)
4427, 42, 43syl2anc 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)
4644, 45syl 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))))
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 13008 . . . . . . 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 7089 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) ∈ {𝑤})
52 elsni 4142 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑤} → (1st𝑧) = 𝑤)
5351, 52syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) = 𝑤)
5453rgen 2906 . . . . . . . . . 10 𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
5554rgenw 2908 . . . . . . . . 9 𝑤 ∈ ((# ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
56 invdisj 4571 . . . . . . . . 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 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))
6159, 60xpeq12d 5064 . . . . . . . . 9 (𝑤 = ((# ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦)))
6261fveq2d 6107 . . . . . . . 8 (𝑤 = ((# ↾ ω)‘𝑦) → (#‘({𝑤} × 𝒫 𝑤)) = (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))))
63 inss2 3796 . . . . . . . . 9 (𝒫 ℕ0 ∩ Fin) ⊆ Fin
6463sseli 3564 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
65 f1of1 6049 . . . . . . . . . 10 ((# ↾ ω):ℕ01-1-onto→ω → (# ↾ ω):ℕ01-1→ω)
6613, 65ax-mp 5 . . . . . . . . 9 (# ↾ ω):ℕ01-1→ω
67 inss1 3795 . . . . . . . . . . 11 (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0
6867sseli 3564 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
6968elpwid 4118 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
70 f1ores 6064 . . . . . . . . 9 (((# ↾ ω):ℕ01-1→ω ∧ 𝑥 ⊆ ℕ0) → ((# ↾ ω) ↾ 𝑥):𝑥1-1-onto→((# ↾ ω) “ 𝑥))
7166, 69, 70sylancr 694 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω) ↾ 𝑥):𝑥1-1-onto→((# ↾ ω) “ 𝑥))
72 fvres 6117 . . . . . . . . 9 (𝑦𝑥 → (((# ↾ ω) ↾ 𝑥)‘𝑦) = ((# ↾ ω)‘𝑦))
7372adantl 481 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (((# ↾ 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)
7641, 74, 753syl 18 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((# ↾ ω) “ 𝑥)) → (#‘({𝑤} × 𝒫 𝑤)) ∈ ℂ)
7762, 64, 71, 73, 76fsumf1o 14301 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ ((# ↾ ω) “ 𝑥)(#‘({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))))
78 snfi 7923 . . . . . . . . . 10 {((# ↾ ω)‘𝑦)} ∈ Fin
7969sselda 3568 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝑦 ∈ ℕ0)
80 f1of 6050 . . . . . . . . . . . . . . 15 ((# ↾ ω):ℕ01-1-onto→ω → (# ↾ ω):ℕ0⟶ω)
8113, 80ax-mp 5 . . . . . . . . . . . . . 14 (# ↾ ω):ℕ0⟶ω
8281ffvelrni 6266 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ0 → ((# ↾ ω)‘𝑦) ∈ ω)
8379, 82syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((# ↾ ω)‘𝑦) ∈ ω)
8434, 83sseldi 3566 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((# ↾ ω)‘𝑦) ∈ Fin)
85 pwfi 8144 . . . . . . . . . . 11 (((# ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 ((# ↾ ω)‘𝑦) ∈ Fin)
8684, 85sylib 207 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 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))))
8878, 86, 87sylancr 694 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))) = ((#‘{((# ↾ ω)‘𝑦)}) · (#‘𝒫 ((# ↾ ω)‘𝑦))))
89 hashsng 13020 . . . . . . . . . . 11 (((# ↾ ω)‘𝑦) ∈ ω → (#‘{((# ↾ ω)‘𝑦)}) = 1)
9083, 89syl 17 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘{((# ↾ ω)‘𝑦)}) = 1)
91 hashpw 13083 . . . . . . . . . . . 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 6117 . . . . . . . . . . . . . 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 6433 . . . . . . . . . . . . . 14 (((# ↾ ω):ω–1-1-onto→ℕ0𝑦 ∈ ℕ0) → ((# ↾ ω)‘((# ↾ ω)‘𝑦)) = 𝑦)
962, 79, 95sylancr 694 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((# ↾ ω)‘((# ↾ ω)‘𝑦)) = 𝑦)
9794, 96eqtr3d 2646 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘((# ↾ ω)‘𝑦)) = 𝑦)
9897oveq2d 6565 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑(#‘((# ↾ ω)‘𝑦))) = (2↑𝑦))
9992, 98eqtrd 2644 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘𝒫 ((# ↾ ω)‘𝑦)) = (2↑𝑦))
10090, 99oveq12d 6567 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((#‘{((# ↾ 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)
103101, 79, 102sylancr 694 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑𝑦) ∈ ℂ)
104103mulid2d 9937 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
10588, 100, 1043eqtrd 2648 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))) = (2↑𝑦))
106105sumeq2dv 14281 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦𝑥 (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))) = Σ𝑦𝑥 (2↑𝑦))
10758, 77, 1063eqtrd 2648 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (#‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (2↑𝑦))
10848, 50, 1073eqtrd 2648 . . . . 5 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω)‘(card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦𝑥 (2↑𝑦))
109108mpteq2ia 4668 . . . 4 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω)‘(card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦𝑥 (2↑𝑦))
11046adantl 481 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
11126adantl 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‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))))
114 iuneq1 4470 . . . . . . . 8 (𝑧 = ((# ↾ ω) “ 𝑥) → 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
115114fveq2d 6107 . . . . . . 7 (𝑧 = ((# ↾ ω) “ 𝑥) → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
116111, 112, 113, 115fmptco 6303 . . . . . 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 6050 . . . . . . . 8 ((# ↾ ω):ω–1-1-onto→ℕ0 → (# ↾ ω):ω⟶ℕ0)
1182, 117mp1i 13 . . . . . . 7 (⊤ → (# ↾ ω):ω⟶ℕ0)
119118feqmptd 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))))
121110, 116, 119, 120fmptco 6303 . . . . 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)))))
122121trud 1484 . . . 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 2642 . . 3 ((# ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥)))) = 𝐹
125 f1oeq1 6040 . . 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 219 1 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ ciun 4455  Disj wdisj 4553   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  dom cdm 5038   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Oncon0 5640  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ωcom 6957  1st c1st 7057  Fincfn 7841  cardccrd 8644  ℂcc 9813  1c1 9816   · cmul 9820  2c2 10947  ℕ0cn0 11169  ↑cexp 12722  #chash 12979  Σcsu 14264 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265 This theorem is referenced by:  bitsinv2  15003
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