Theorem ackbijnn 15753

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 14303	. . . 4 ⊢ (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
2	1	hashgf1o 13896	. . 3 ⊢ (♯ ↾ ω):ω–1-1-onto→ℕ0
3		sneq 4589	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
4		pweq 4567	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
5	3, 4	xpeq12d 5654	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
6	5	cbviunv 4992	. . . . . . . 8 ⊢ ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦)
7		iuneq1 4961	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
8	6, 7	eqtrid 2776	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
9	8	fveq2d 6830	. . . . . 6 ⊢ (𝑧 = 𝑥 → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
10	9	cbvmptv 5199	. . . . 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 6780	. . . . . 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 9265	. . . . 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 6791	. . . 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 692	. . 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 6791	. . 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 692	. 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 4191	. . . . . . . . . 10 ⊢ (𝒫 ω ∩ Fin) ⊆ Fin
21		f1of 6768	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥))
24	23	fmpt 7048	. . . . . . . . . . . 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 3220	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
27	20, 26	sselid 3935	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(♯ ↾ ω) “ 𝑥) ∈ Fin)
28		snfi 8975	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
29		cnvimass 6037	. . . . . . . . . . . . . . 15 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ dom (♯ ↾ ω)
30		dmhashres 14266	. . . . . . . . . . . . . . 15 ⊢ dom (♯ ↾ ω) = ω
31	29, 30	sseqtri 3986	. . . . . . . . . . . . . 14 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ ω
32		onfin2 9140	. . . . . . . . . . . . . . 15 ⊢ ω = (On ∩ Fin)
33		inss2 4191	. . . . . . . . . . . . . . 15 ⊢ (On ∩ Fin) ⊆ Fin
34	32, 33	eqsstri 3984	. . . . . . . . . . . . . 14 ⊢ ω ⊆ Fin
35	31, 34	sstri 3947	. . . . . . . . . . . . 13 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ Fin
36		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥))
37	35, 36	sselid 3935	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38		pwfi 9226	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
39	37, 38	sylib 218	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40		xpfi 9227	. . . . . . . . . . 11 ⊢ (({𝑤} ∈ Fin ∧ 𝒫 𝑤 ∈ Fin) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
41	28, 39, 40	sylancr 587	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
42	41	ralrimiva 3121	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43		iunfi 9252	. . . . . . . . 9 ⊢ (((◡(♯ ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → ∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
44	27, 42, 43	syl2anc 584	. . . . . . . 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 6846	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
48		hashcard 14280	. . . . . . 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 7963	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) ∈ {𝑤})
51		elsni 4596	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ {𝑤} → (1st ‘𝑧) = 𝑤)
52	50, 51	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) = 𝑤)
53	52	rgen 3046	. . . . . . . . . 10 ⊢ ∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
54	53	rgenw 3048	. . . . . . . . 9 ⊢ ∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
55		invdisj 5081	. . . . . . . . 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 15747	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)))
58		sneq 4589	. . . . . . . . . 10 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → {𝑤} = {(◡(♯ ↾ ω)‘𝑦)})
59		pweq 4567	. . . . . . . . . 10 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 (◡(♯ ↾ ω)‘𝑦))
60	58, 59	xpeq12d 5654	. . . . . . . . 9 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦)))
61	60	fveq2d 6830	. . . . . . . 8 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → (♯‘({𝑤} × 𝒫 𝑤)) = (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))))
62		elinel2 4155	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
63		f1of1 6767	. . . . . . . . . 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 4154	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
66	65	elpwid 4562	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
67		f1ores 6782	. . . . . . . . 9 ⊢ ((◡(♯ ↾ ω):ℕ0–1-1→ω ∧ 𝑥 ⊆ ℕ0) → (◡(♯ ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(♯ ↾ ω) “ 𝑥))
68	64, 66, 67	sylancr 587	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(♯ ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(♯ ↾ ω) “ 𝑥))
69		fvres 6845	. . . . . . . . 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 14281	. . . . . . . . 9 ⊢ (({𝑤} × 𝒫 𝑤) ∈ Fin → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
72		nn0cn 12412	. . . . . . . . 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 15648	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))))
75		snfi 8975	. . . . . . . . . 10 ⊢ {(◡(♯ ↾ ω)‘𝑦)} ∈ Fin
76	66	sselda 3937	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℕ0)
77		f1of 6768	. . . . . . . . . . . . . . 15 ⊢ (◡(♯ ↾ ω):ℕ0–1-1-onto→ω → ◡(♯ ↾ ω):ℕ0⟶ω)
78	13, 77	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ◡(♯ ↾ ω):ℕ0⟶ω
79	78	ffvelcdmi 7021	. . . . . . . . . . . . 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 3935	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (◡(♯ ↾ ω)‘𝑦) ∈ Fin)
82		pwfi 9226	. . . . . . . . . . 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 14359	. . . . . . . . . 10 ⊢ (({(◡(♯ ↾ ω)‘𝑦)} ∈ Fin ∧ 𝒫 (◡(♯ ↾ ω)‘𝑦) ∈ Fin) → (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = ((♯‘{(◡(♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦))))
85	75, 83, 84	sylancr 587	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = ((♯‘{(◡(♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦))))
86		hashsng 14294	. . . . . . . . . . 11 ⊢ ((◡(♯ ↾ ω)‘𝑦) ∈ ω → (♯‘{(◡(♯ ↾ ω)‘𝑦)}) = 1)
87	80, 86	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘{(◡(♯ ↾ ω)‘𝑦)}) = 1)
88		hashpw 14361	. . . . . . . . . . . 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 6846	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((♯ ↾ ω)‘(◡(♯ ↾ ω)‘𝑦)) = (♯‘(◡(♯ ↾ ω)‘𝑦)))
91		f1ocnvfv2 7218	. . . . . . . . . . . . . 14 ⊢ (((♯ ↾ ω):ω–1-1-onto→ℕ0 ∧ 𝑦 ∈ ℕ0) → ((♯ ↾ ω)‘(◡(♯ ↾ ω)‘𝑦)) = 𝑦)
92	2, 76, 91	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((♯ ↾ ω)‘(◡(♯ ↾ ω)‘𝑦)) = 𝑦)
93	90, 92	eqtr3d 2766	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘(◡(♯ ↾ ω)‘𝑦)) = 𝑦)
94	93	oveq2d 7369	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑(♯‘(◡(♯ ↾ ω)‘𝑦))) = (2↑𝑦))
95	89, 94	eqtrd 2764	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦)) = (2↑𝑦))
96	87, 95	oveq12d 7371	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((♯‘{(◡(♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
97		2cn 12221	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
98		expcl 14004	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
99	97, 76, 98	sylancr 587	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑𝑦) ∈ ℂ)
100	99	mullidd 11152	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
101	85, 96, 100	3eqtrd 2768	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = (2↑𝑦))
102	101	sumeq2dv 15627	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦 ∈ 𝑥 (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
103	57, 74, 102	3eqtrd 2768	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
104	47, 49, 103	3eqtrd 2768	. . . . 5 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
105	104	mpteq2ia 5190	. . . 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 2730	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))
109		eqidd 2730	. . . . . . 7 ⊢ (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))))
110		iuneq1 4961	. . . . . . . 8 ⊢ (𝑧 = (◡(♯ ↾ ω) “ 𝑥) → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
111	110	fveq2d 6830	. . . . . . 7 ⊢ (𝑧 = (◡(♯ ↾ ω) “ 𝑥) → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
112	107, 108, 109, 111	fmptco 7067	. . . . . 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 6768	. . . . . . . 8 ⊢ ((♯ ↾ ω):ω–1-1-onto→ℕ0 → (♯ ↾ ω):ω⟶ℕ0)
114	2, 113	mp1i 13	. . . . . . 7 ⊢ (⊤ → (♯ ↾ ω):ω⟶ℕ0)
115	114	feqmptd 6895	. . . . . 6 ⊢ (⊤ → (♯ ↾ ω) = (𝑦 ∈ ω ↦ ((♯ ↾ ω)‘𝑦)))
116		fveq2 6826	. . . . . 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 7067	. . . . 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 1547	. . . 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 2762	. . 3 ⊢ ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))) = 𝐹
121		f1oeq1 6756	. . 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 1540  ⊤wtru 1541   ∈ wcel 2109  ∀wral 3044   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4553  {csn 4579  ∪ ciun 4944  Disj wdisj 5062   ↦ cmpt 5176   × cxp 5621  ^◡ccnv 5622  dom cdm 5623   ↾ cres 5625   “ cima 5626   ∘ ccom 5627  Oncon0 6311  ⟶wf 6482  –1-1→wf1 6483  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  ωcom 7806  1^st c1st 7929  Fincfn 8879  cardccrd 9850  ℂcc 11026  1c1 11029   · cmul 11033  2c2 12201  ℕ₀cn0 12402  ↑cexp 13986  ♯chash 14255  Σcsu 15611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-disj 5063  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8632  df-map 8762  df-pm 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-oi 9421  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-xnn0 12476  df-z 12490  df-uz 12754  df-rp 12912  df-fz 13429  df-fzo 13576  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-clim 15413  df-sum 15612

This theorem is referenced by:  bitsinv2  16372