Theorem ackbijnn 15540

Description: Translate the Ackermann bijection ackbij1 9994 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 14093	. . . 4 ⊢ (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
2	1	hashgf1o 13691	. . 3 ⊢ (♯ ↾ ω):ω–1-1-onto→ℕ0
3		sneq 4571	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
4		pweq 4549	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
5	3, 4	xpeq12d 5620	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
6	5	cbviunv 4970	. . . . . . . 8 ⊢ ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦)
7		iuneq1 4940	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
8	6, 7	eqtrid 2790	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
9	8	fveq2d 6778	. . . . . 6 ⊢ (𝑧 = 𝑥 → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
10	9	cbvmptv 5187	. . . . 5 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
11	10	ackbij1 9994	. . . 4 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12		f1ocnv 6728	. . . . . 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 9123	. . . . 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 6739	. . . 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 689	. . 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 6739	. . 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 689	. 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 4163	. . . . . . . . . 10 ⊢ (𝒫 ω ∩ Fin) ⊆ Fin
21		f1of 6716	. . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥))
24	23	fmpt 6984	. . . . . . . . . . . 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 230	. . . . . . . . . . 11 ⊢ ∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)(◡(♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin)
26	25	rspec 3133	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
27	20, 26	sselid 3919	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(♯ ↾ ω) “ 𝑥) ∈ Fin)
28		snfi 8834	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
29		cnvimass 5989	. . . . . . . . . . . . . . 15 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ dom (♯ ↾ ω)
30		dmhashres 14055	. . . . . . . . . . . . . . 15 ⊢ dom (♯ ↾ ω) = ω
31	29, 30	sseqtri 3957	. . . . . . . . . . . . . 14 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ ω
32		onfin2 9014	. . . . . . . . . . . . . . 15 ⊢ ω = (On ∩ Fin)
33		inss2 4163	. . . . . . . . . . . . . . 15 ⊢ (On ∩ Fin) ⊆ Fin
34	32, 33	eqsstri 3955	. . . . . . . . . . . . . 14 ⊢ ω ⊆ Fin
35	31, 34	sstri 3930	. . . . . . . . . . . . 13 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ Fin
36		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥))
37	35, 36	sselid 3919	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38		pwfi 8961	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
39	37, 38	sylib 217	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40		xpfi 9085	. . . . . . . . . . 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 3103	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43		iunfi 9107	. . . . . . . . 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 9719	. . . . . . . 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 6794	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
48		hashcard 14070	. . . . . . 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 7863	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) ∈ {𝑤})
51		elsni 4578	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ {𝑤} → (1st ‘𝑧) = 𝑤)
52	50, 51	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) = 𝑤)
53	52	rgen 3074	. . . . . . . . . 10 ⊢ ∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
54	53	rgenw 3076	. . . . . . . . 9 ⊢ ∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
55		invdisj 5058	. . . . . . . . 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 15534	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)))
58		sneq 4571	. . . . . . . . . 10 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → {𝑤} = {(◡(♯ ↾ ω)‘𝑦)})
59		pweq 4549	. . . . . . . . . 10 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 (◡(♯ ↾ ω)‘𝑦))
60	58, 59	xpeq12d 5620	. . . . . . . . 9 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦)))
61	60	fveq2d 6778	. . . . . . . 8 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → (♯‘({𝑤} × 𝒫 𝑤)) = (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))))
62		elinel2 4130	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
63		f1of1 6715	. . . . . . . . . 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 4129	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
66	65	elpwid 4544	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
67		f1ores 6730	. . . . . . . . 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 6793	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → ((◡(♯ ↾ ω) ↾ 𝑥)‘𝑦) = (◡(♯ ↾ ω)‘𝑦))
70	69	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((◡(♯ ↾ ω) ↾ 𝑥)‘𝑦) = (◡(♯ ↾ ω)‘𝑦))
71		hashcl 14071	. . . . . . . . 9 ⊢ (({𝑤} × 𝒫 𝑤) ∈ Fin → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
72		nn0cn 12243	. . . . . . . . 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 15435	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))))
75		snfi 8834	. . . . . . . . . 10 ⊢ {(◡(♯ ↾ ω)‘𝑦)} ∈ Fin
76	66	sselda 3921	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℕ0)
77		f1of 6716	. . . . . . . . . . . . . . 15 ⊢ (◡(♯ ↾ ω):ℕ0–1-1-onto→ω → ◡(♯ ↾ ω):ℕ0⟶ω)
78	13, 77	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ◡(♯ ↾ ω):ℕ0⟶ω
79	78	ffvelrni 6960	. . . . . . . . . . . . 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 3919	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (◡(♯ ↾ ω)‘𝑦) ∈ Fin)
82		pwfi 8961	. . . . . . . . . . 11 ⊢ ((◡(♯ ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 (◡(♯ ↾ ω)‘𝑦) ∈ Fin)
83	81, 82	sylib 217	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝒫 (◡(♯ ↾ ω)‘𝑦) ∈ Fin)
84		hashxp 14149	. . . . . . . . . 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 14084	. . . . . . . . . . 11 ⊢ ((◡(♯ ↾ ω)‘𝑦) ∈ ω → (♯‘{(◡(♯ ↾ ω)‘𝑦)}) = 1)
87	80, 86	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘{(◡(♯ ↾ ω)‘𝑦)}) = 1)
88		hashpw 14151	. . . . . . . . . . . 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 6794	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((♯ ↾ ω)‘(◡(♯ ↾ ω)‘𝑦)) = (♯‘(◡(♯ ↾ ω)‘𝑦)))
91		f1ocnvfv2 7149	. . . . . . . . . . . . . 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 2780	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘(◡(♯ ↾ ω)‘𝑦)) = 𝑦)
94	93	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑(♯‘(◡(♯ ↾ ω)‘𝑦))) = (2↑𝑦))
95	89, 94	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦)) = (2↑𝑦))
96	87, 95	oveq12d 7293	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((♯‘{(◡(♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
97		2cn 12048	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
98		expcl 13800	. . . . . . . . . . 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	mulid2d 10993	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
101	85, 96, 100	3eqtrd 2782	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = (2↑𝑦))
102	101	sumeq2dv 15415	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦 ∈ 𝑥 (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
103	57, 74, 102	3eqtrd 2782	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
104	47, 49, 103	3eqtrd 2782	. . . . 5 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
105	104	mpteq2ia 5177	. . . 4 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (2↑𝑦))
106	46	adantl 482	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
107	26	adantl 482	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (◡(♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
108		eqidd 2739	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))
109		eqidd 2739	. . . . . . 7 ⊢ (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))))
110		iuneq1 4940	. . . . . . . 8 ⊢ (𝑧 = (◡(♯ ↾ ω) “ 𝑥) → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
111	110	fveq2d 6778	. . . . . . 7 ⊢ (𝑧 = (◡(♯ ↾ ω) “ 𝑥) → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
112	107, 108, 109, 111	fmptco 7001	. . . . . 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 6716	. . . . . . . 8 ⊢ ((♯ ↾ ω):ω–1-1-onto→ℕ0 → (♯ ↾ ω):ω⟶ℕ0)
114	2, 113	mp1i 13	. . . . . . 7 ⊢ (⊤ → (♯ ↾ ω):ω⟶ℕ0)
115	114	feqmptd 6837	. . . . . 6 ⊢ (⊤ → (♯ ↾ ω) = (𝑦 ∈ ω ↦ ((♯ ↾ ω)‘𝑦)))
116		fveq2 6774	. . . . . 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 7001	. . . . 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 1546	. . . 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 2776	. . 3 ⊢ ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))) = 𝐹
121		f1oeq1 6704	. . 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 229	1 ⊢ 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ⊤wtru 1540   ∈ wcel 2106  ∀wral 3064   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561  ∪ ciun 4924  Disj wdisj 5039   ↦ cmpt 5157   × cxp 5587  ^◡ccnv 5588  dom cdm 5589   ↾ cres 5591   “ cima 5592   ∘ ccom 5593  Oncon0 6266  ⟶wf 6429  –1-1→wf1 6430  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275  ωcom 7712  1^st c1st 7829  Fincfn 8733  cardccrd 9693  ℂcc 10869  1c1 10872   · cmul 10876  2c2 12028  ℕ₀cn0 12233  ↑cexp 13782  ♯chash 14044  Σcsu 15397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398

This theorem is referenced by:  bitsinv2  16150