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Theorem ackbijnn 14604
Description: Translate the Ackermann bijection ackbij1 9098 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 13205 . . . 4 (# ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
21hashgf1o 12810 . . 3 (# ↾ ω):ω–1-1-onto→ℕ0
3 sneq 4220 . . . . . . . . . 10 (𝑤 = 𝑦 → {𝑤} = {𝑦})
4 pweq 4194 . . . . . . . . . 10 (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
53, 4xpeq12d 5174 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
65cbviunv 4591 . . . . . . . 8 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑧 ({𝑦} × 𝒫 𝑦)
7 iuneq1 4566 . . . . . . . 8 (𝑧 = 𝑥 𝑦𝑧 ({𝑦} × 𝒫 𝑦) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
86, 7syl5eq 2697 . . . . . . 7 (𝑧 = 𝑥 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
98fveq2d 6233 . . . . . 6 (𝑧 = 𝑥 → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
109cbvmptv 4783 . . . . 5 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
1110ackbij1 9098 . . . 4 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12 f1ocnv 6187 . . . . . 6 ((# ↾ ω):ω–1-1-onto→ℕ0(# ↾ ω):ℕ01-1-onto→ω)
132, 12ax-mp 5 . . . . 5 (# ↾ ω):ℕ01-1-onto→ω
14 f1opwfi 8311 . . . . 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 6197 . . . 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 708 . . 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 6197 . . 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 708 . 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 3867 . . . . . . . . . 10 (𝒫 ω ∩ Fin) ⊆ Fin
21 f1of 6175 . . . . . . . . . . . . 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 2651 . . . . . . . . . . . . 13 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥))
2423fmpt 6421 . . . . . . . . . . . 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 2960 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
2720, 26sseldi 3634 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω) “ 𝑥) ∈ Fin)
28 snfi 8079 . . . . . . . . . . 11 {𝑤} ∈ Fin
29 cnvimass 5520 . . . . . . . . . . . . . . 15 ((# ↾ ω) “ 𝑥) ⊆ dom (# ↾ ω)
30 dmhashres 13169 . . . . . . . . . . . . . . 15 dom (# ↾ ω) = ω
3129, 30sseqtri 3670 . . . . . . . . . . . . . 14 ((# ↾ ω) “ 𝑥) ⊆ ω
32 onfin2 8193 . . . . . . . . . . . . . . 15 ω = (On ∩ Fin)
33 inss2 3867 . . . . . . . . . . . . . . 15 (On ∩ Fin) ⊆ Fin
3432, 33eqsstri 3668 . . . . . . . . . . . . . 14 ω ⊆ Fin
3531, 34sstri 3645 . . . . . . . . . . . . 13 ((# ↾ ω) “ 𝑥) ⊆ Fin
36 simpr 476 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((# ↾ ω) “ 𝑥)) → 𝑤 ∈ ((# ↾ ω) “ 𝑥))
3735, 36sseldi 3634 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((# ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38 pwfi 8302 . . . . . . . . . . . 12 (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
3937, 38sylib 208 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((# ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40 xpfi 8272 . . . . . . . . . . 11 (({𝑤} ∈ Fin ∧ 𝒫 𝑤 ∈ Fin) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
4128, 39, 40sylancr 696 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((# ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
4241ralrimiva 2995 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43 iunfi 8295 . . . . . . . . 9 ((((# ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
4427, 42, 43syl2anc 694 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
45 ficardom 8825 . . . . . . . 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 6245 . . . . . . 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 13184 . . . . . . 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 7242 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) ∈ {𝑤})
52 elsni 4227 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑤} → (1st𝑧) = 𝑤)
5351, 52syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) = 𝑤)
5453rgen 2951 . . . . . . . . . 10 𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
5554rgenw 2953 . . . . . . . . 9 𝑤 ∈ ((# ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
56 invdisj 4670 . . . . . . . . 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 14598 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (#‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ ((# ↾ ω) “ 𝑥)(#‘({𝑤} × 𝒫 𝑤)))
59 sneq 4220 . . . . . . . . . 10 (𝑤 = ((# ↾ ω)‘𝑦) → {𝑤} = {((# ↾ ω)‘𝑦)})
60 pweq 4194 . . . . . . . . . 10 (𝑤 = ((# ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 ((# ↾ ω)‘𝑦))
6159, 60xpeq12d 5174 . . . . . . . . 9 (𝑤 = ((# ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦)))
6261fveq2d 6233 . . . . . . . 8 (𝑤 = ((# ↾ ω)‘𝑦) → (#‘({𝑤} × 𝒫 𝑤)) = (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))))
63 inss2 3867 . . . . . . . . 9 (𝒫 ℕ0 ∩ Fin) ⊆ Fin
6463sseli 3632 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
65 f1of1 6174 . . . . . . . . . 10 ((# ↾ ω):ℕ01-1-onto→ω → (# ↾ ω):ℕ01-1→ω)
6613, 65ax-mp 5 . . . . . . . . 9 (# ↾ ω):ℕ01-1→ω
67 inss1 3866 . . . . . . . . . . 11 (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0
6867sseli 3632 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
6968elpwid 4203 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
70 f1ores 6189 . . . . . . . . 9 (((# ↾ ω):ℕ01-1→ω ∧ 𝑥 ⊆ ℕ0) → ((# ↾ ω) ↾ 𝑥):𝑥1-1-onto→((# ↾ ω) “ 𝑥))
7166, 69, 70sylancr 696 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω) ↾ 𝑥):𝑥1-1-onto→((# ↾ ω) “ 𝑥))
72 fvres 6245 . . . . . . . . 9 (𝑦𝑥 → (((# ↾ ω) ↾ 𝑥)‘𝑦) = ((# ↾ ω)‘𝑦))
7372adantl 481 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (((# ↾ ω) ↾ 𝑥)‘𝑦) = ((# ↾ ω)‘𝑦))
74 hashcl 13185 . . . . . . . . 9 (({𝑤} × 𝒫 𝑤) ∈ Fin → (#‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
75 nn0cn 11340 . . . . . . . . 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 14498 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ ((# ↾ ω) “ 𝑥)(#‘({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))))
78 snfi 8079 . . . . . . . . . 10 {((# ↾ ω)‘𝑦)} ∈ Fin
7969sselda 3636 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝑦 ∈ ℕ0)
80 f1of 6175 . . . . . . . . . . . . . . 15 ((# ↾ ω):ℕ01-1-onto→ω → (# ↾ ω):ℕ0⟶ω)
8113, 80ax-mp 5 . . . . . . . . . . . . . 14 (# ↾ ω):ℕ0⟶ω
8281ffvelrni 6398 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ0 → ((# ↾ ω)‘𝑦) ∈ ω)
8379, 82syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((# ↾ ω)‘𝑦) ∈ ω)
8434, 83sseldi 3634 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((# ↾ ω)‘𝑦) ∈ Fin)
85 pwfi 8302 . . . . . . . . . . 11 (((# ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 ((# ↾ ω)‘𝑦) ∈ Fin)
8684, 85sylib 208 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝒫 ((# ↾ ω)‘𝑦) ∈ Fin)
87 hashxp 13259 . . . . . . . . . 10 (({((# ↾ ω)‘𝑦)} ∈ Fin ∧ 𝒫 ((# ↾ ω)‘𝑦) ∈ Fin) → (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))) = ((#‘{((# ↾ ω)‘𝑦)}) · (#‘𝒫 ((# ↾ ω)‘𝑦))))
8878, 86, 87sylancr 696 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))) = ((#‘{((# ↾ ω)‘𝑦)}) · (#‘𝒫 ((# ↾ ω)‘𝑦))))
89 hashsng 13197 . . . . . . . . . . 11 (((# ↾ ω)‘𝑦) ∈ ω → (#‘{((# ↾ ω)‘𝑦)}) = 1)
9083, 89syl 17 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘{((# ↾ ω)‘𝑦)}) = 1)
91 hashpw 13261 . . . . . . . . . . . 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 6245 . . . . . . . . . . . . . 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 6573 . . . . . . . . . . . . . 14 (((# ↾ ω):ω–1-1-onto→ℕ0𝑦 ∈ ℕ0) → ((# ↾ ω)‘((# ↾ ω)‘𝑦)) = 𝑦)
962, 79, 95sylancr 696 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((# ↾ ω)‘((# ↾ ω)‘𝑦)) = 𝑦)
9794, 96eqtr3d 2687 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘((# ↾ ω)‘𝑦)) = 𝑦)
9897oveq2d 6706 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑(#‘((# ↾ ω)‘𝑦))) = (2↑𝑦))
9992, 98eqtrd 2685 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘𝒫 ((# ↾ ω)‘𝑦)) = (2↑𝑦))
10090, 99oveq12d 6708 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((#‘{((# ↾ ω)‘𝑦)}) · (#‘𝒫 ((# ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
101 2cn 11129 . . . . . . . . . . 11 2 ∈ ℂ
102 expcl 12918 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
103101, 79, 102sylancr 696 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑𝑦) ∈ ℂ)
104103mulid2d 10096 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
10588, 100, 1043eqtrd 2689 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))) = (2↑𝑦))
106105sumeq2dv 14477 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦𝑥 (#‘({((# ↾ ω)‘𝑦)} × 𝒫 ((# ↾ ω)‘𝑦))) = Σ𝑦𝑥 (2↑𝑦))
10758, 77, 1063eqtrd 2689 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (#‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (2↑𝑦))
10848, 50, 1073eqtrd 2689 . . . . 5 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω)‘(card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦𝑥 (2↑𝑦))
109108mpteq2ia 4773 . . . 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 2652 . . . . . . 7 (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥)))
113 eqidd 2652 . . . . . . 7 (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))))
114 iuneq1 4566 . . . . . . . 8 (𝑧 = ((# ↾ ω) “ 𝑥) → 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
115114fveq2d 6233 . . . . . . 7 (𝑧 = ((# ↾ ω) “ 𝑥) → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
116111, 112, 113, 115fmptco 6436 . . . . . 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 6175 . . . . . . . 8 ((# ↾ ω):ω–1-1-onto→ℕ0 → (# ↾ ω):ω⟶ℕ0)
1182, 117mp1i 13 . . . . . . 7 (⊤ → (# ↾ ω):ω⟶ℕ0)
119118feqmptd 6288 . . . . . 6 (⊤ → (# ↾ ω) = (𝑦 ∈ ω ↦ ((# ↾ ω)‘𝑦)))
120 fveq2 6229 . . . . . 6 (𝑦 = (card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) → ((# ↾ ω)‘𝑦) = ((# ↾ ω)‘(card‘ 𝑤 ∈ ((# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
121110, 116, 119, 120fmptco 6436 . . . . 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 1533 . . . 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 2683 . . 3 ((# ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω) “ 𝑥)))) = 𝐹
125 f1oeq1 6165 . . 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 383   = wceq 1523  wtru 1524  wcel 2030  wral 2941  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210   ciun 4552  Disj wdisj 4652  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  cres 5145  cima 5146  ccom 5147  Oncon0 5761  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  ωcom 7107  1st c1st 7208  Fincfn 7997  cardccrd 8799  cc 9972  1c1 9975   · cmul 9979  2c2 11108  0cn0 11330  cexp 12900  #chash 13157  Σcsu 14460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461
This theorem is referenced by:  bitsinv2  15212
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