Theorem ackbijnn 15784

Description: Translate the Ackermann bijection ackbij1 10150 onto the positive integers. (Contributed by Mario Carneiro, 16-Jan-2015.)

Hypothesis

Ref	Expression
ackbijnn.1	⊢ 𝐹 = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (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.

Step	Hyp	Ref	Expression
1		hashgval2 14331	. . . 4 ⊢ (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
2	1	hashgf1o 13924	. . 3 ⊢ (♯ ↾ ω):ω–1-1-onto→ℕ0
3		sneq 4578	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
4		pweq 4556	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
5	3, 4	xpeq12d 5655	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
6	5	cbviunv 4982	. . . . . . . 8 ⊢ ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦)
7		iuneq1 4951	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
8	6, 7	eqtrid 2784	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
9	8	fveq2d 6838	. . . . . 6 ⊢ (𝑧 = 𝑥 → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
10	9	cbvmptv 5190	. . . . 5 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
11	10	ackbij1 10150	. . . 4 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12		f1ocnv 6786	. . . . . 6 ⊢ ((♯ ↾ ω):ω–1-1-onto→ℕ0 → ◡(♯ ↾ ω):ℕ0–1-1-onto→ω)
13	2, 12	ax-mp 5	. . . . 5 ⊢ ◡(♯ ↾ ω):ℕ0–1-1-onto→ω
14		f1opwfi 9259	. . . . 5 ⊢ (◡(♯ ↾ ω):ℕ0–1-1-onto→ω → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)–1-1-onto→(𝒫 ω ∩ Fin))
15	13, 14	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)–1-1-onto→(𝒫 ω ∩ Fin)
16		f1oco 6797	. . . 4 ⊢ (((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({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‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ω)
17	11, 15, 16	mp2an 693	. . 3 ⊢ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ω
18		f1oco 6797	. . 3 ⊢ (((♯ ↾ ω):ω–1-1-onto→ℕ0 ∧ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ω) → ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0)
19	2, 17, 18	mp2an 693	. 2 ⊢ ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0
20		inss2 4179	. . . . . . . . . 10 ⊢ (𝒫 ω ∩ Fin) ⊆ Fin
21		f1of 6774	. . . . . . . . . . . . 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))
22	15, 21	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)⟶(𝒫 ω ∩ Fin)
23		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥))
24	23	fmpt 7056	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)(◡(♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin) ↔ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)):(𝒫 ℕ0 ∩ Fin)⟶(𝒫 ω ∩ Fin))
25	22, 24	mpbir 231	. . . . . . . . . . 11 ⊢ ∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)(◡(♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin)
26	25	rspec 3229	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
27	20, 26	sselid 3920	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(♯ ↾ ω) “ 𝑥) ∈ Fin)
28		snfi 8983	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
29		cnvimass 6041	. . . . . . . . . . . . . . 15 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ dom (♯ ↾ ω)
30		dmhashres 14294	. . . . . . . . . . . . . . 15 ⊢ dom (♯ ↾ ω) = ω
31	29, 30	sseqtri 3971	. . . . . . . . . . . . . 14 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ ω
32		onfin2 9144	. . . . . . . . . . . . . . 15 ⊢ ω = (On ∩ Fin)
33		inss2 4179	. . . . . . . . . . . . . . 15 ⊢ (On ∩ Fin) ⊆ Fin
34	32, 33	eqsstri 3969	. . . . . . . . . . . . . 14 ⊢ ω ⊆ Fin
35	31, 34	sstri 3932	. . . . . . . . . . . . 13 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ Fin
36		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥))
37	35, 36	sselid 3920	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38		pwfi 9222	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
39	37, 38	sylib 218	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40		xpfi 9223	. . . . . . . . . . 11 ⊢ (({𝑤} ∈ Fin ∧ 𝒫 𝑤 ∈ Fin) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
41	28, 39, 40	sylancr 588	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
42	41	ralrimiva 3130	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43		iunfi 9246	. . . . . . . . 9 ⊢ (((◡(♯ ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → ∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
44	27, 42, 43	syl2anc 585	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
45		ficardom 9876	. . . . . . . 8 ⊢ (∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin → (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
46	44, 45	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
47	46	fvresd 6854	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
48		hashcard 14308	. . . . . . 7 ⊢ (∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin → (♯‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
49	44, 48	syl 17	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
50		xp1st 7967	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) ∈ {𝑤})
51		elsni 4585	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ {𝑤} → (1st ‘𝑧) = 𝑤)
52	50, 51	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) = 𝑤)
53	52	rgen 3054	. . . . . . . . . 10 ⊢ ∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
54	53	rgenw 3056	. . . . . . . . 9 ⊢ ∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
55		invdisj 5072	. . . . . . . . 9 ⊢ (∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤 → Disj 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
56	54, 55	mp1i 13	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Disj 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
57	27, 41, 56	hashiun 15776	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)))
58		sneq 4578	. . . . . . . . . 10 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → {𝑤} = {(◡(♯ ↾ ω)‘𝑦)})
59		pweq 4556	. . . . . . . . . 10 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 (◡(♯ ↾ ω)‘𝑦))
60	58, 59	xpeq12d 5655	. . . . . . . . 9 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦)))
61	60	fveq2d 6838	. . . . . . . 8 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → (♯‘({𝑤} × 𝒫 𝑤)) = (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))))
62		elinel2 4143	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
63		f1of1 6773	. . . . . . . . . 10 ⊢ (◡(♯ ↾ ω):ℕ0–1-1-onto→ω → ◡(♯ ↾ ω):ℕ0–1-1→ω)
64	13, 63	ax-mp 5	. . . . . . . . 9 ⊢ ◡(♯ ↾ ω):ℕ0–1-1→ω
65		elinel1 4142	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
66	65	elpwid 4551	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
67		f1ores 6788	. . . . . . . . 9 ⊢ ((◡(♯ ↾ ω):ℕ0–1-1→ω ∧ 𝑥 ⊆ ℕ0) → (◡(♯ ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(♯ ↾ ω) “ 𝑥))
68	64, 66, 67	sylancr 588	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(♯ ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(♯ ↾ ω) “ 𝑥))
69		fvres 6853	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → ((◡(♯ ↾ ω) ↾ 𝑥)‘𝑦) = (◡(♯ ↾ ω)‘𝑦))
70	69	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((◡(♯ ↾ ω) ↾ 𝑥)‘𝑦) = (◡(♯ ↾ ω)‘𝑦))
71		hashcl 14309	. . . . . . . . 9 ⊢ (({𝑤} × 𝒫 𝑤) ∈ Fin → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
72		nn0cn 12438	. . . . . . . . 9 ⊢ ((♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0 → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℂ)
73	41, 71, 72	3syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℂ)
74	61, 62, 68, 70, 73	fsumf1o 15676	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))))
75		snfi 8983	. . . . . . . . . 10 ⊢ {(◡(♯ ↾ ω)‘𝑦)} ∈ Fin
76	66	sselda 3922	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℕ0)
77		f1of 6774	. . . . . . . . . . . . . . 15 ⊢ (◡(♯ ↾ ω):ℕ0–1-1-onto→ω → ◡(♯ ↾ ω):ℕ0⟶ω)
78	13, 77	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ◡(♯ ↾ ω):ℕ0⟶ω
79	78	ffvelcdmi 7029	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0 → (◡(♯ ↾ ω)‘𝑦) ∈ ω)
80	76, 79	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (◡(♯ ↾ ω)‘𝑦) ∈ ω)
81	34, 80	sselid 3920	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (◡(♯ ↾ ω)‘𝑦) ∈ Fin)
82		pwfi 9222	. . . . . . . . . . 11 ⊢ ((◡(♯ ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 (◡(♯ ↾ ω)‘𝑦) ∈ Fin)
83	81, 82	sylib 218	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝒫 (◡(♯ ↾ ω)‘𝑦) ∈ Fin)
84		hashxp 14387	. . . . . . . . . 10 ⊢ (({(◡(♯ ↾ ω)‘𝑦)} ∈ Fin ∧ 𝒫 (◡(♯ ↾ ω)‘𝑦) ∈ Fin) → (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = ((♯‘{(◡(♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦))))
85	75, 83, 84	sylancr 588	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = ((♯‘{(◡(♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦))))
86		hashsng 14322	. . . . . . . . . . 11 ⊢ ((◡(♯ ↾ ω)‘𝑦) ∈ ω → (♯‘{(◡(♯ ↾ ω)‘𝑦)}) = 1)
87	80, 86	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘{(◡(♯ ↾ ω)‘𝑦)}) = 1)
88		hashpw 14389	. . . . . . . . . . . 12 ⊢ ((◡(♯ ↾ ω)‘𝑦) ∈ Fin → (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦)) = (2↑(♯‘(◡(♯ ↾ ω)‘𝑦))))
89	81, 88	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦)) = (2↑(♯‘(◡(♯ ↾ ω)‘𝑦))))
90	80	fvresd 6854	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((♯ ↾ ω)‘(◡(♯ ↾ ω)‘𝑦)) = (♯‘(◡(♯ ↾ ω)‘𝑦)))
91		f1ocnvfv2 7225	. . . . . . . . . . . . . 14 ⊢ (((♯ ↾ ω):ω–1-1-onto→ℕ0 ∧ 𝑦 ∈ ℕ0) → ((♯ ↾ ω)‘(◡(♯ ↾ ω)‘𝑦)) = 𝑦)
92	2, 76, 91	sylancr 588	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((♯ ↾ ω)‘(◡(♯ ↾ ω)‘𝑦)) = 𝑦)
93	90, 92	eqtr3d 2774	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘(◡(♯ ↾ ω)‘𝑦)) = 𝑦)
94	93	oveq2d 7376	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑(♯‘(◡(♯ ↾ ω)‘𝑦))) = (2↑𝑦))
95	89, 94	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦)) = (2↑𝑦))
96	87, 95	oveq12d 7378	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((♯‘{(◡(♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
97		2cn 12247	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
98		expcl 14032	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
99	97, 76, 98	sylancr 588	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑𝑦) ∈ ℂ)
100	99	mullidd 11154	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
101	85, 96, 100	3eqtrd 2776	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = (2↑𝑦))
102	101	sumeq2dv 15655	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦 ∈ 𝑥 (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
103	57, 74, 102	3eqtrd 2776	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
104	47, 49, 103	3eqtrd 2776	. . . . 5 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
105	104	mpteq2ia 5181	. . . 4 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (2↑𝑦))
106	46	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
107	26	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (◡(♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
108		eqidd 2738	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))
109		eqidd 2738	. . . . . . 7 ⊢ (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))))
110		iuneq1 4951	. . . . . . . 8 ⊢ (𝑧 = (◡(♯ ↾ ω) “ 𝑥) → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
111	110	fveq2d 6838	. . . . . . 7 ⊢ (𝑧 = (◡(♯ ↾ ω) “ 𝑥) → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
112	107, 108, 109, 111	fmptco 7076	. . . . . 6 ⊢ (⊤ → ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
113		f1of 6774	. . . . . . . 8 ⊢ ((♯ ↾ ω):ω–1-1-onto→ℕ0 → (♯ ↾ ω):ω⟶ℕ0)
114	2, 113	mp1i 13	. . . . . . 7 ⊢ (⊤ → (♯ ↾ ω):ω⟶ℕ0)
115	114	feqmptd 6902	. . . . . 6 ⊢ (⊤ → (♯ ↾ ω) = (𝑦 ∈ ω ↦ ((♯ ↾ ω)‘𝑦)))
116		fveq2 6834	. . . . . 6 ⊢ (𝑦 = (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) → ((♯ ↾ ω)‘𝑦) = ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
117	106, 112, 115, 116	fmptco 7076	. . . . 5 ⊢ (⊤ → ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))))
118	117	mptru 1549	. . . 4 ⊢ ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({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) ↦ Σ𝑦 ∈ 𝑥 (2↑𝑦))
120	105, 118, 119	3eqtr4i 2770	. . 3 ⊢ ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))) = 𝐹
121		f1oeq1 6762	. . 3 ⊢ (((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))) = 𝐹 → (((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0 ↔ 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0))
122	120, 121	ax-mp 5	. 2 ⊢ (((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))):(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0 ↔ 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0)
123	19, 122	mpbi 230	1 ⊢ 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  ∀wral 3052   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  {csn 4568  ∪ ciun 4934  Disj wdisj 5053   ↦ cmpt 5167   × cxp 5622  ^◡ccnv 5623  dom cdm 5624   ↾ cres 5626   “ cima 5627   ∘ ccom 5628  Oncon0 6317  ⟶wf 6488  –1-1→wf1 6489  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7360  ωcom 7810  1^st c1st 7933  Fincfn 8886  cardccrd 9850  ℂcc 11027  1c1 11030   · cmul 11034  2c2 12227  ℕ₀cn0 12428  ↑cexp 14014  ♯chash 14283  Σcsu 15639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-disj 5054  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-er 8636  df-map 8768  df-pm 8769  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-oi 9418  df-dju 9816  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-xnn0 12502  df-z 12516  df-uz 12780  df-rp 12934  df-fz 13453  df-fzo 13600  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-sum 15640

This theorem is referenced by:  bitsinv2  16403