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Theorem ackbijnn 14485
Description: Translate the Ackermann bijection ackbij1 9004 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 13107 . . . 4 (# ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
21hashgf1o 12710 . . 3 (# ↾ ω):ω–1-1-onto→ℕ0
3 sneq 4158 . . . . . . . . . 10 (𝑤 = 𝑦 → {𝑤} = {𝑦})
4 pweq 4133 . . . . . . . . . 10 (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
53, 4xpeq12d 5100 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
65cbviunv 4525 . . . . . . . 8 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑧 ({𝑦} × 𝒫 𝑦)
7 iuneq1 4500 . . . . . . . 8 (𝑧 = 𝑥 𝑦𝑧 ({𝑦} × 𝒫 𝑦) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
86, 7syl5eq 2667 . . . . . . 7 (𝑧 = 𝑥 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
98fveq2d 6152 . . . . . 6 (𝑧 = 𝑥 → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
109cbvmptv 4710 . . . . 5 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
1110ackbij1 9004 . . . 4 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12 f1ocnv 6106 . . . . . 6 ((# ↾ ω):ω–1-1-onto→ℕ0(# ↾ ω):ℕ01-1-onto→ω)
132, 12ax-mp 5 . . . . 5 (# ↾ ω):ℕ01-1-onto→ω
14 f1opwfi 8214 . . . . 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 6116 . . . 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 707 . . 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 6116 . . 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 707 . 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 3812 . . . . . . . . . 10 (𝒫 ω ∩ Fin) ⊆ Fin
21 f1of 6094 . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . 13 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥))
2423fmpt 6337 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin) ↔ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)⟶(𝒫 ω ∩ Fin))
2522, 24mpbir 221 . . . . . . . . . . 11 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin)
2625rspec 2926 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
2720, 26sseldi 3581 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω) “ 𝑥) ∈ Fin)
28 snfi 7982 . . . . . . . . . . 11 {𝑤} ∈ Fin
29 cnvimass 5444 . . . . . . . . . . . . . . 15 ((# ↾ ω) “ 𝑥) ⊆ dom (# ↾ ω)
30 dmhashres 13069 . . . . . . . . . . . . . . 15 dom (# ↾ ω) = ω
3129, 30sseqtri 3616 . . . . . . . . . . . . . 14 ((# ↾ ω) “ 𝑥) ⊆ ω
32 onfin2 8096 . . . . . . . . . . . . . . 15 ω = (On ∩ Fin)
33 inss2 3812 . . . . . . . . . . . . . . 15 (On ∩ Fin) ⊆ Fin
3432, 33eqsstri 3614 . . . . . . . . . . . . . 14 ω ⊆ Fin
3531, 34sstri 3592 . . . . . . . . . . . . 13 ((# ↾ ω) “ 𝑥) ⊆ Fin
36 simpr 477 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((# ↾ ω) “ 𝑥)) → 𝑤 ∈ ((# ↾ ω) “ 𝑥))
3735, 36sseldi 3581 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((# ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38 pwfi 8205 . . . . . . . . . . . 12 (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
3937, 38sylib 208 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((# ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40 xpfi 8175 . . . . . . . . . . 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 2960 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43 iunfi 8198 . . . . . . . . 9 ((((# ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
4427, 42, 43syl2anc 692 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
45 ficardom 8731 . . . . . . . 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 6164 . . . . . . 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 13086 . . . . . . 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 7143 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) ∈ {𝑤})
52 elsni 4165 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑤} → (1st𝑧) = 𝑤)
5351, 52syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) = 𝑤)
5453rgen 2917 . . . . . . . . . 10 𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
5554rgenw 2919 . . . . . . . . 9 𝑤 ∈ ((# ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
56 invdisj 4601 . . . . . . . . 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 14481 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (#‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ ((# ↾ ω) “ 𝑥)(#‘({𝑤} × 𝒫 𝑤)))
59 sneq 4158 . . . . . . . . . 10 (𝑤 = ((# ↾ ω)‘𝑦) → {𝑤} = {((# ↾ ω)‘𝑦)})
60 pweq 4133 . . . . . . . . . 10 (𝑤 = ((# ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 ((# ↾ ω)‘𝑦))
6159, 60xpeq12d 5100 . . . . . . . . 9 (𝑤 = ((# ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦)))
6261fveq2d 6152 . . . . . . . 8 (𝑤 = ((# ↾ ω)‘𝑦) → (#‘({𝑤} × 𝒫 𝑤)) = (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))))
63 inss2 3812 . . . . . . . . 9 (𝒫 ℕ0 ∩ Fin) ⊆ Fin
6463sseli 3579 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
65 f1of1 6093 . . . . . . . . . 10 ((# ↾ ω):ℕ01-1-onto→ω → (# ↾ ω):ℕ01-1→ω)
6613, 65ax-mp 5 . . . . . . . . 9 (# ↾ ω):ℕ01-1→ω
67 inss1 3811 . . . . . . . . . . 11 (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0
6867sseli 3579 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
6968elpwid 4141 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
70 f1ores 6108 . . . . . . . . 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 6164 . . . . . . . . 9 (𝑦𝑥 → (((# ↾ ω) ↾ 𝑥)‘𝑦) = ((# ↾ ω)‘𝑦))
7372adantl 482 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (((# ↾ ω) ↾ 𝑥)‘𝑦) = ((# ↾ ω)‘𝑦))
74 hashcl 13087 . . . . . . . . 9 (({𝑤} × 𝒫 𝑤) ∈ Fin → (#‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
75 nn0cn 11246 . . . . . . . . 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 14387 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ ((# ↾ ω) “ 𝑥)(#‘({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))))
78 snfi 7982 . . . . . . . . . 10 {((# ↾ ω)‘𝑦)} ∈ Fin
7969sselda 3583 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝑦 ∈ ℕ0)
80 f1of 6094 . . . . . . . . . . . . . . 15 ((# ↾ ω):ℕ01-1-onto→ω → (# ↾ ω):ℕ0⟶ω)
8113, 80ax-mp 5 . . . . . . . . . . . . . 14 (# ↾ ω):ℕ0⟶ω
8281ffvelrni 6314 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ0 → ((# ↾ ω)‘𝑦) ∈ ω)
8379, 82syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((# ↾ ω)‘𝑦) ∈ ω)
8434, 83sseldi 3581 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((# ↾ ω)‘𝑦) ∈ Fin)
85 pwfi 8205 . . . . . . . . . . 11 (((# ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 ((# ↾ ω)‘𝑦) ∈ Fin)
8684, 85sylib 208 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝒫 ((# ↾ ω)‘𝑦) ∈ Fin)
87 hashxp 13161 . . . . . . . . . 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 13099 . . . . . . . . . . 11 (((# ↾ ω)‘𝑦) ∈ ω → (#‘{((# ↾ ω)‘𝑦)}) = 1)
9083, 89syl 17 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘{((# ↾ ω)‘𝑦)}) = 1)
91 hashpw 13163 . . . . . . . . . . . 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 6164 . . . . . . . . . . . . . 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 6487 . . . . . . . . . . . . . 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 2657 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘((# ↾ ω)‘𝑦)) = 𝑦)
9897oveq2d 6620 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑(#‘((# ↾ ω)‘𝑦))) = (2↑𝑦))
9992, 98eqtrd 2655 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘𝒫 ((# ↾ ω)‘𝑦)) = (2↑𝑦))
10090, 99oveq12d 6622 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((#‘{((# ↾ ω)‘𝑦)}) · (#‘𝒫 ((# ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
101 2cn 11035 . . . . . . . . . . 11 2 ∈ ℂ
102 expcl 12818 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
103101, 79, 102sylancr 694 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑𝑦) ∈ ℂ)
104103mulid2d 10002 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
10588, 100, 1043eqtrd 2659 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))) = (2↑𝑦))
106105sumeq2dv 14367 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦𝑥 (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))) = Σ𝑦𝑥 (2↑𝑦))
10758, 77, 1063eqtrd 2659 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (#‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (2↑𝑦))
10848, 50, 1073eqtrd 2659 . . . . 5 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω)‘(card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦𝑥 (2↑𝑦))
109108mpteq2ia 4700 . . . 4 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω)‘(card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦𝑥 (2↑𝑦))
11046adantl 482 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
11126adantl 482 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → ((# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
112 eqidd 2622 . . . . . . 7 (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥)))
113 eqidd 2622 . . . . . . 7 (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))))
114 iuneq1 4500 . . . . . . . 8 (𝑧 = ((# ↾ ω) “ 𝑥) → 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
115114fveq2d 6152 . . . . . . 7 (𝑧 = ((# ↾ ω) “ 𝑥) → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
116111, 112, 113, 115fmptco 6351 . . . . . 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 6094 . . . . . . . 8 ((# ↾ ω):ω–1-1-onto→ℕ0 → (# ↾ ω):ω⟶ℕ0)
1182, 117mp1i 13 . . . . . . 7 (⊤ → (# ↾ ω):ω⟶ℕ0)
119118feqmptd 6206 . . . . . 6 (⊤ → (# ↾ ω) = (𝑦 ∈ ω ↦ ((# ↾ ω)‘𝑦)))
120 fveq2 6148 . . . . . 6 (𝑦 = (card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) → ((# ↾ ω)‘𝑦) = ((# ↾ ω)‘(card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
121110, 116, 119, 120fmptco 6351 . . . . 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 1490 . . . 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 2653 . . 3 ((# ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥)))) = 𝐹
125 f1oeq1 6084 . . 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 220 1 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wtru 1481  wcel 1987  wral 2907  cin 3554  wss 3555  𝒫 cpw 4130  {csn 4148   ciun 4485  Disj wdisj 4583  cmpt 4673   × cxp 5072  ccnv 5073  dom cdm 5074  cres 5076  cima 5077  ccom 5078  Oncon0 5682  wf 5843  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  ωcom 7012  1st c1st 7111  Fincfn 7899  cardccrd 8705  cc 9878  1c1 9881   · cmul 9885  2c2 11014  0cn0 11236  cexp 12800  #chash 13057  Σcsu 14350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351
This theorem is referenced by:  bitsinv2  15089
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