Theorem ackbijnn 16003

Description: Translate the Ackermann bijection ackbij1 10319 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 14526	. . . 4 ⊢ (♯ ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
2	1	hashgf1o 14118	. . 3 ⊢ (♯ ↾ ω):ω–1-1-onto→ℕ0
3		sneq 4590	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
4		pweq 4570	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
5	3, 4	xpeq12d 5681	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
6	5	cbviunv 4996	. . . . . . . 8 ⊢ ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦)
7		iuneq1 4967	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
8	6, 7	eqtrid 2807	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
9	8	fveq2d 6890	. . . . . 6 ⊢ (𝑧 = 𝑥 → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
10	9	cbvmptv 5208	. . . . 5 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
11	10	ackbij1 10319	. . . 4 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12		f1ocnv 6838	. . . . . 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 9349	. . . . 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 6849	. . . 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 705	. . 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 6849	. . 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 705	. 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 4134	. . . . . . . . . 10 ⊢ (𝒫 ω ∩ Fin) ⊆ Fin
21		f1of 6825	. . . . . . . . . . . . 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 2760	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥))
24	23	fmpt 7111	. . . . . . . . . . . 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 234	. . . . . . . . . . 11 ⊢ ∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)(◡(♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin)
26	25	rspec 3253	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
27	20, 26	sselid 3906	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(♯ ↾ ω) “ 𝑥) ∈ Fin)
28		snfi 9075	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
29		cnvimass 6198	. . . . . . . . . . . . . . 15 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ dom (♯ ↾ ω)
30		dmhashres 14489	. . . . . . . . . . . . . . 15 ⊢ dom (♯ ↾ ω) = ω
31	29, 30	sseqtri 3956	. . . . . . . . . . . . . 14 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ ω
32		onfin2 9236	. . . . . . . . . . . . . . 15 ⊢ ω = (On ∩ Fin)
33		inss2 4134	. . . . . . . . . . . . . . 15 ⊢ (On ∩ Fin) ⊆ Fin
34	32, 33	eqsstri 3954	. . . . . . . . . . . . . 14 ⊢ ω ⊆ Fin
35	31, 34	sstri 3917	. . . . . . . . . . . . 13 ⊢ (◡(♯ ↾ ω) “ 𝑥) ⊆ Fin
36		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥))
37	35, 36	sselid 3906	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38		pwfi 9314	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
39	37, 38	sylib 221	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40		xpfi 9315	. . . . . . . . . . 11 ⊢ (({𝑤} ∈ Fin ∧ 𝒫 𝑤 ∈ Fin) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
41	28, 39, 40	sylancr 599	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
42	41	ralrimiva 3154	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43		iunfi 9336	. . . . . . . . 9 ⊢ (((◡(♯ ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → ∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
44	27, 42, 43	syl2anc 596	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
45		ficardom 10046	. . . . . . . 8 ⊢ (∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin → (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
46	44, 45	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
47	46	fvresd 6906	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
48		hashcard 14503	. . . . . . 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 18	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (♯‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
50		xp1st 8034	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) ∈ {𝑤})
51		elsni 4597	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ {𝑤} → (1st ‘𝑧) = 𝑤)
52	50, 51	syl 18	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) = 𝑤)
53	52	rgen 3078	. . . . . . . . . 10 ⊢ ∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
54	53	rgenw 3080	. . . . . . . . 9 ⊢ ∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
55		invdisj 5088	. . . . . . . . 9 ⊢ (∀𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤 → Disj 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
56	54, 55	mp1i 14	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Disj 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
57	27, 41, 56	hashiun 15995	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)))
58		sneq 4590	. . . . . . . . . 10 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → {𝑤} = {(◡(♯ ↾ ω)‘𝑦)})
59		pweq 4570	. . . . . . . . . 10 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 (◡(♯ ↾ ω)‘𝑦))
60	58, 59	xpeq12d 5681	. . . . . . . . 9 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦)))
61	60	fveq2d 6890	. . . . . . . 8 ⊢ (𝑤 = (◡(♯ ↾ ω)‘𝑦) → (♯‘({𝑤} × 𝒫 𝑤)) = (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))))
62		elinel2 4099	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
63		f1of1 6824	. . . . . . . . . 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 4098	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
66	65	elpwid 4565	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
67		f1ores 6840	. . . . . . . . 9 ⊢ ((◡(♯ ↾ ω):ℕ0–1-1→ω ∧ 𝑥 ⊆ ℕ0) → (◡(♯ ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(♯ ↾ ω) “ 𝑥))
68	64, 66, 67	sylancr 599	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(♯ ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(♯ ↾ ω) “ 𝑥))
69		fvres 6905	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → ((◡(♯ ↾ ω) ↾ 𝑥)‘𝑦) = (◡(♯ ↾ ω)‘𝑦))
70	69	adantl 487	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((◡(♯ ↾ ω) ↾ 𝑥)‘𝑦) = (◡(♯ ↾ ω)‘𝑦))
71		hashcl 14504	. . . . . . . . 9 ⊢ (({𝑤} × 𝒫 𝑤) ∈ Fin → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
72		nn0cn 12620	. . . . . . . . 9 ⊢ ((♯‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0 → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℂ)
73	41, 71, 72	3syl 19	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)) → (♯‘({𝑤} × 𝒫 𝑤)) ∈ ℂ)
74	61, 62, 68, 70, 73	fsumf1o 15895	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)(♯‘({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))))
75		snfi 9075	. . . . . . . . . 10 ⊢ {(◡(♯ ↾ ω)‘𝑦)} ∈ Fin
76	66	sselda 3908	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℕ0)
77		f1of 6825	. . . . . . . . . . . . . . 15 ⊢ (◡(♯ ↾ ω):ℕ0–1-1-onto→ω → ◡(♯ ↾ ω):ℕ0⟶ω)
78	13, 77	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ◡(♯ ↾ ω):ℕ0⟶ω
79	78	ffvelcdmi 7084	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0 → (◡(♯ ↾ ω)‘𝑦) ∈ ω)
80	76, 79	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (◡(♯ ↾ ω)‘𝑦) ∈ ω)
81	34, 80	sselid 3906	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (◡(♯ ↾ ω)‘𝑦) ∈ Fin)
82		pwfi 9314	. . . . . . . . . . 11 ⊢ ((◡(♯ ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 (◡(♯ ↾ ω)‘𝑦) ∈ Fin)
83	81, 82	sylib 221	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝒫 (◡(♯ ↾ ω)‘𝑦) ∈ Fin)
84		hashxp 14583	. . . . . . . . . 10 ⊢ (({(◡(♯ ↾ ω)‘𝑦)} ∈ Fin ∧ 𝒫 (◡(♯ ↾ ω)‘𝑦) ∈ Fin) → (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = ((♯‘{(◡(♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦))))
85	75, 83, 84	sylancr 599	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = ((♯‘{(◡(♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦))))
86		hashsng 14517	. . . . . . . . . . 11 ⊢ ((◡(♯ ↾ ω)‘𝑦) ∈ ω → (♯‘{(◡(♯ ↾ ω)‘𝑦)}) = 1)
87	80, 86	syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘{(◡(♯ ↾ ω)‘𝑦)}) = 1)
88		hashpw 14585	. . . . . . . . . . . 12 ⊢ ((◡(♯ ↾ ω)‘𝑦) ∈ Fin → (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦)) = (2↑(♯‘(◡(♯ ↾ ω)‘𝑦))))
89	81, 88	syl 18	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦)) = (2↑(♯‘(◡(♯ ↾ ω)‘𝑦))))
90	80	fvresd 6906	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((♯ ↾ ω)‘(◡(♯ ↾ ω)‘𝑦)) = (♯‘(◡(♯ ↾ ω)‘𝑦)))
91		f1ocnvfv2 7286	. . . . . . . . . . . . . 14 ⊢ (((♯ ↾ ω):ω–1-1-onto→ℕ0 ∧ 𝑦 ∈ ℕ0) → ((♯ ↾ ω)‘(◡(♯ ↾ ω)‘𝑦)) = 𝑦)
92	2, 76, 91	sylancr 599	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((♯ ↾ ω)‘(◡(♯ ↾ ω)‘𝑦)) = 𝑦)
93	90, 92	eqtr3d 2797	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘(◡(♯ ↾ ω)‘𝑦)) = 𝑦)
94	93	oveq2d 7437	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑(♯‘(◡(♯ ↾ ω)‘𝑦))) = (2↑𝑦))
95	89, 94	eqtrd 2795	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦)) = (2↑𝑦))
96	87, 95	oveq12d 7439	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((♯‘{(◡(♯ ↾ ω)‘𝑦)}) · (♯‘𝒫 (◡(♯ ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
97		2cn 12422	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
98		expcl 14226	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
99	97, 76, 98	sylancr 599	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑𝑦) ∈ ℂ)
100	99	mullidd 11331	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
101	85, 96, 100	3eqtrd 2799	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = (2↑𝑦))
102	101	sumeq2dv 15875	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦 ∈ 𝑥 (♯‘({(◡(♯ ↾ ω)‘𝑦)} × 𝒫 (◡(♯ ↾ ω)‘𝑦))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
103	57, 74, 102	3eqtrd 2799	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (♯‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
104	47, 49, 103	3eqtrd 2799	. . . . 5 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
105	104	mpteq2ia 5199	. . . 4 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((♯ ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (2↑𝑦))
106	46	adantl 487	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
107	26	adantl 487	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (◡(♯ ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
108		eqidd 2761	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))
109		eqidd 2761	. . . . . . 7 ⊢ (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))))
110		iuneq1 4967	. . . . . . . 8 ⊢ (𝑧 = (◡(♯ ↾ ω) “ 𝑥) → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
111	110	fveq2d 6890	. . . . . . 7 ⊢ (𝑧 = (◡(♯ ↾ ω) “ 𝑥) → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑤 ∈ (◡(♯ ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
112	107, 108, 109, 111	fmptco 7131	. . . . . 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 6825	. . . . . . . 8 ⊢ ((♯ ↾ ω):ω–1-1-onto→ℕ0 → (♯ ↾ ω):ω⟶ℕ0)
114	2, 113	mp1i 14	. . . . . . 7 ⊢ (⊤ → (♯ ↾ ω):ω⟶ℕ0)
115	114	feqmptd 6954	. . . . . 6 ⊢ (⊤ → (♯ ↾ ω) = (𝑦 ∈ ω ↦ ((♯ ↾ ω)‘𝑦)))
116		fveq2 6886	. . . . . 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 7131	. . . . 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 1576	. . . 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 2793	. . 3 ⊢ ((♯ ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(♯ ↾ ω) “ 𝑥)))) = 𝐹
121		f1oeq1 6813	. . 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 233	1 ⊢ 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ⊤wtru 1570   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556  {csn 4584  ∪ ciun 4950  Disj wdisj 5069   ↦ cmpt 5185   × cxp 5648  ^◡ccnv 5649  dom cdm 5650   ↾ cres 5652   “ cima 5653   ∘ ccom 5654  Oncon0 6362  ⟶wf 6535  –1-1→wf1 6536  –1-1-onto→wf1o 6538  ‘cfv 6539  (class class class)co 7421  ωcom 7878  1^st c1st 8000  Fincfn 8977  cardccrd 10020  ℂcc 11202  1c1 11205   · cmul 11209  2c2 12401  ℕ₀cn0 12610  ↑cexp 14208  ♯chash 14478  Σcsu 15859

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-inf2 9646  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281  ax-pre-sup 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8721  df-map 8853  df-pm 8854  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-sup 9438  df-oi 9508  df-dju 9986  df-card 10024  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-div 11978  df-nn 12340  df-2 12409  df-3 12410  df-n0 12611  df-xnn0 12684  df-z 12698  df-uz 12970  df-rp 13125  df-fz 13644  df-fzo 13793  df-seq 14149  df-exp 14209  df-hash 14479  df-cj 15272  df-re 15273  df-im 15274  df-sqrt 15408  df-abs 15409  df-clim 15661  df-sum 15860

This theorem is used by:  bitsinv2  16619