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Theorem ackbijnn 15832
Description: Translate the Ackermann bijection ackbij1 10281 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 14395 . . . 4 (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
21hashgf1o 13991 . . 3 (♯ ↾ ω):ω–1-1-onto→ℕ0
3 sneq 4643 . . . . . . . . . 10 (𝑤 = 𝑦 → {𝑤} = {𝑦})
4 pweq 4621 . . . . . . . . . 10 (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
53, 4xpeq12d 5713 . . . . . . . . 9 (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
65cbviunv 5048 . . . . . . . 8 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑧 ({𝑦} × 𝒫 𝑦)
7 iuneq1 5017 . . . . . . . 8 (𝑧 = 𝑥 𝑦𝑧 ({𝑦} × 𝒫 𝑦) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
86, 7eqtrid 2778 . . . . . . 7 (𝑧 = 𝑥 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑦𝑥 ({𝑦} × 𝒫 𝑦))
98fveq2d 6905 . . . . . 6 (𝑧 = 𝑥 → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
109cbvmptv 5266 . . . . 5 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
1110ackbij1 10281 . . . 4 (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12 f1ocnv 6855 . . . . . 6 ((♯ ↾ ω):ω–1-1-onto→ℕ0(♯ ↾ ω):ℕ01-1-onto→ω)
132, 12ax-mp 5 . . . . 5 (♯ ↾ ω):ℕ01-1-onto→ω
14 f1opwfi 9400 . . . . 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 6866 . . . 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 690 . . 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 6866 . . 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 690 . 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 4231 . . . . . . . . . 10 (𝒫 ω ∩ Fin) ⊆ Fin
21 f1of 6843 . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . 13 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥))
2423fmpt 7124 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin) ↔ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)⟶(𝒫 ω ∩ Fin))
2522, 24mpbir 230 . . . . . . . . . . 11 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin)
2625rspec 3238 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
2720, 26sselid 3977 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) “ 𝑥) ∈ Fin)
28 snfi 9081 . . . . . . . . . . 11 {𝑤} ∈ Fin
29 cnvimass 6091 . . . . . . . . . . . . . . 15 ((♯ ↾ ω) “ 𝑥) ⊆ dom (♯ ↾ ω)
30 dmhashres 14358 . . . . . . . . . . . . . . 15 dom (♯ ↾ ω) = ω
3129, 30sseqtri 4016 . . . . . . . . . . . . . 14 ((♯ ↾ ω) “ 𝑥) ⊆ ω
32 onfin2 9265 . . . . . . . . . . . . . . 15 ω = (On ∩ Fin)
33 inss2 4231 . . . . . . . . . . . . . . 15 (On ∩ Fin) ⊆ Fin
3432, 33eqsstri 4014 . . . . . . . . . . . . . 14 ω ⊆ Fin
3531, 34sstri 3989 . . . . . . . . . . . . 13 ((♯ ↾ ω) “ 𝑥) ⊆ Fin
36 simpr 483 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥))
3735, 36sselid 3977 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38 pwfi 9359 . . . . . . . . . . . 12 (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
3937, 38sylib 217 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40 xpfi 9360 . . . . . . . . . . 11 (({𝑤} ∈ Fin ∧ 𝒫 𝑤 ∈ Fin) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
4128, 39, 40sylancr 585 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
4241ralrimiva 3136 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43 iunfi 9385 . . . . . . . . 9 ((((♯ ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
4427, 42, 43syl2anc 582 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
45 ficardom 10004 . . . . . . . 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 6921 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
48 hashcard 14372 . . . . . . 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 8035 . . . . . . . . . . . 12 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) ∈ {𝑤})
51 elsni 4650 . . . . . . . . . . . 12 ((1st𝑧) ∈ {𝑤} → (1st𝑧) = 𝑤)
5250, 51syl 17 . . . . . . . . . . 11 (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st𝑧) = 𝑤)
5352rgen 3053 . . . . . . . . . 10 𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
5453rgenw 3055 . . . . . . . . 9 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st𝑧) = 𝑤
55 invdisj 5137 . . . . . . . . 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 15826 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ ((♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)))
58 sneq 4643 . . . . . . . . . 10 (𝑤 = ((♯ ↾ ω)‘𝑦) → {𝑤} = {((♯ ↾ ω)‘𝑦)})
59 pweq 4621 . . . . . . . . . 10 (𝑤 = ((♯ ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 ((♯ ↾ ω)‘𝑦))
6058, 59xpeq12d 5713 . . . . . . . . 9 (𝑤 = ((♯ ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦)))
6160fveq2d 6905 . . . . . . . 8 (𝑤 = ((♯ ↾ ω)‘𝑦) → (♯‘({𝑤} × 𝒫 𝑤)) = (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))))
62 elinel2 4197 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
63 f1of1 6842 . . . . . . . . . 10 ((♯ ↾ ω):ℕ01-1-onto→ω → (♯ ↾ ω):ℕ01-1→ω)
6413, 63ax-mp 5 . . . . . . . . 9 (♯ ↾ ω):ℕ01-1→ω
65 elinel1 4196 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
6665elpwid 4616 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
67 f1ores 6857 . . . . . . . . 9 (((♯ ↾ ω):ℕ01-1→ω ∧ 𝑥 ⊆ ℕ0) → ((♯ ↾ ω) ↾ 𝑥):𝑥1-1-onto→((♯ ↾ ω) “ 𝑥))
6864, 66, 67sylancr 585 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω) ↾ 𝑥):𝑥1-1-onto→((♯ ↾ ω) “ 𝑥))
69 fvres 6920 . . . . . . . . 9 (𝑦𝑥 → (((♯ ↾ ω) ↾ 𝑥)‘𝑦) = ((♯ ↾ ω)‘𝑦))
7069adantl 480 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (((♯ ↾ ω) ↾ 𝑥)‘𝑦) = ((♯ ↾ ω)‘𝑦))
71 hashcl 14373 . . . . . . . . 9 (({𝑤} × 𝒫 𝑤) ∈ Fin → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
72 nn0cn 12534 . . . . . . . . 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 15727 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ ((♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))))
75 snfi 9081 . . . . . . . . . 10 {((♯ ↾ ω)‘𝑦)} ∈ Fin
7666sselda 3979 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝑦 ∈ ℕ0)
77 f1of 6843 . . . . . . . . . . . . . . 15 ((♯ ↾ ω):ℕ01-1-onto→ω → (♯ ↾ ω):ℕ0⟶ω)
7813, 77ax-mp 5 . . . . . . . . . . . . . 14 (♯ ↾ ω):ℕ0⟶ω
7978ffvelcdmi 7097 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ0 → ((♯ ↾ ω)‘𝑦) ∈ ω)
8076, 79syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘𝑦) ∈ ω)
8134, 80sselid 3977 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘𝑦) ∈ Fin)
82 pwfi 9359 . . . . . . . . . . 11 (((♯ ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin)
8381, 82sylib 217 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin)
84 hashxp 14451 . . . . . . . . . 10 (({((♯ ↾ ω)‘𝑦)} ∈ Fin ∧ 𝒫 ((♯ ↾ ω)‘𝑦) ∈ Fin) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))))
8575, 83, 84sylancr 585 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))))
86 hashsng 14386 . . . . . . . . . . 11 (((♯ ↾ ω)‘𝑦) ∈ ω → (♯‘{((♯ ↾ ω)‘𝑦)}) = 1)
8780, 86syl 17 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘{((♯ ↾ ω)‘𝑦)}) = 1)
88 hashpw 14453 . . . . . . . . . . . 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 6921 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = (♯‘((♯ ↾ ω)‘𝑦)))
91 f1ocnvfv2 7291 . . . . . . . . . . . . . 14 (((♯ ↾ ω):ω–1-1-onto→ℕ0𝑦 ∈ ℕ0) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = 𝑦)
922, 76, 91sylancr 585 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯ ↾ ω)‘((♯ ↾ ω)‘𝑦)) = 𝑦)
9390, 92eqtr3d 2768 . . . . . . . . . . . 12 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘((♯ ↾ ω)‘𝑦)) = 𝑦)
9493oveq2d 7440 . . . . . . . . . . 11 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑(♯‘((♯ ↾ ω)‘𝑦))) = (2↑𝑦))
9589, 94eqtrd 2766 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘𝒫 ((♯ ↾ ω)‘𝑦)) = (2↑𝑦))
9687, 95oveq12d 7442 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → ((♯‘{((♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 ((♯ ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
97 2cn 12339 . . . . . . . . . . 11 2 ∈ ℂ
98 expcl 14099 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
9997, 76, 98sylancr 585 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (2↑𝑦) ∈ ℂ)
10099mullidd 11282 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
10185, 96, 1003eqtrd 2770 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦𝑥) → (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = (2↑𝑦))
102101sumeq2dv 15707 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦𝑥 (♯‘({((♯ ↾ ω)‘𝑦)} × 𝒫 ((♯ ↾ ω)‘𝑦))) = Σ𝑦𝑥 (2↑𝑦))
10357, 74, 1023eqtrd 2770 . . . . . 6 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦𝑥 (2↑𝑦))
10447, 49, 1033eqtrd 2770 . . . . 5 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦𝑥 (2↑𝑦))
105104mpteq2ia 5256 . . . 4 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦𝑥 (2↑𝑦))
10646adantl 480 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
10726adantl 480 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → ((♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
108 eqidd 2727 . . . . . . 7 (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))
109 eqidd 2727 . . . . . . 7 (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))))
110 iuneq1 5017 . . . . . . . 8 (𝑧 = ((♯ ↾ ω) “ 𝑥) → 𝑤𝑧 ({𝑤} × 𝒫 𝑤) = 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
111110fveq2d 6905 . . . . . . 7 (𝑧 = ((♯ ↾ ω) “ 𝑥) → (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
112107, 108, 109, 111fmptco 7143 . . . . . 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 6843 . . . . . . . 8 ((♯ ↾ ω):ω–1-1-onto→ℕ0 → (♯ ↾ ω):ω⟶ℕ0)
1142, 113mp1i 13 . . . . . . 7 (⊤ → (♯ ↾ ω):ω⟶ℕ0)
115114feqmptd 6971 . . . . . 6 (⊤ → (♯ ↾ ω) = (𝑦 ∈ ω ↦ ((♯ ↾ ω)‘𝑦)))
116 fveq2 6901 . . . . . 6 (𝑦 = (card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) → ((♯ ↾ ω)‘𝑦) = ((♯ ↾ ω)‘(card‘ 𝑤 ∈ ((♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
117106, 112, 115, 116fmptco 7143 . . . . 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 1541 . . . 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 2764 . . 3 ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑤𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω) “ 𝑥)))) = 𝐹
121 f1oeq1 6831 . . 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 229 1 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1534  wtru 1535  wcel 2099  wral 3051  cin 3946  wss 3947  𝒫 cpw 4607  {csn 4633   ciun 5001  Disj wdisj 5118  cmpt 5236   × cxp 5680  ccnv 5681  dom cdm 5682  cres 5684  cima 5685  ccom 5686  Oncon0 6376  wf 6550  1-1wf1 6551  1-1-ontowf1o 6553  cfv 6554  (class class class)co 7424  ωcom 7876  1st c1st 8001  Fincfn 8974  cardccrd 9978  cc 11156  1c1 11159   · cmul 11163  2c2 12319  0cn0 12524  cexp 14081  chash 14347  Σcsu 15690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-disj 5119  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-oadd 8500  df-er 8734  df-map 8857  df-pm 8858  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-oi 9553  df-dju 9944  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12597  df-z 12611  df-uz 12875  df-rp 13029  df-fz 13539  df-fzo 13682  df-seq 14022  df-exp 14082  df-hash 14348  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-clim 15490  df-sum 15691
This theorem is referenced by:  bitsinv2  16443
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