Theorem ackbijnn 14399

Description: Translate the Ackermann bijection ackbij1 8943 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 13028	. . . 4 ⊢ (# ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
2	1	hashgf1o 12632	. . 3 ⊢ (# ↾ ω):ω–1-1-onto→ℕ0
3		sneq 4135	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
4		pweq 4111	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
5	3, 4	xpeq12d 5064	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
6	5	cbviunv 4495	. . . . . . . 8 ⊢ ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦)
7		iuneq1 4470	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
8	6, 7	syl5eq 2656	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
9	8	fveq2d 6107	. . . . . 6 ⊢ (𝑧 = 𝑥 → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
10	9	cbvmptv 4678	. . . . 5 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
11	10	ackbij1 8943	. . . 4 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12		f1ocnv 6062	. . . . . 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 8153	. . . . 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 6072	. . . 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 704	. . 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 6072	. . 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 704	. 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 3796	. . . . . . . . . 10 ⊢ (𝒫 ω ∩ Fin) ⊆ Fin
21		f1of 6050	. . . . . . . . . . . . 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 2610	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥))
24	23	fmpt 6289	. . . . . . . . . . . 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 220	. . . . . . . . . . 11 ⊢ ∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)(◡(# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin)
26	25	rspec 2915	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
27	20, 26	sseldi 3566	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(# ↾ ω) “ 𝑥) ∈ Fin)
28		snfi 7923	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
29		cnvimass 5404	. . . . . . . . . . . . . . 15 ⊢ (◡(# ↾ ω) “ 𝑥) ⊆ dom (# ↾ ω)
30		dmhashres 12991	. . . . . . . . . . . . . . 15 ⊢ dom (# ↾ ω) = ω
31	29, 30	sseqtri 3600	. . . . . . . . . . . . . 14 ⊢ (◡(# ↾ ω) “ 𝑥) ⊆ ω
32		onfin2 8037	. . . . . . . . . . . . . . 15 ⊢ ω = (On ∩ Fin)
33		inss2 3796	. . . . . . . . . . . . . . 15 ⊢ (On ∩ Fin) ⊆ Fin
34	32, 33	eqsstri 3598	. . . . . . . . . . . . . 14 ⊢ ω ⊆ Fin
35	31, 34	sstri 3577	. . . . . . . . . . . . 13 ⊢ (◡(# ↾ ω) “ 𝑥) ⊆ Fin
36		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → 𝑤 ∈ (◡(# ↾ ω) “ 𝑥))
37	35, 36	sseldi 3566	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38		pwfi 8144	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
39	37, 38	sylib 207	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40		xpfi 8116	. . . . . . . . . . 11 ⊢ (({𝑤} ∈ Fin ∧ 𝒫 𝑤 ∈ Fin) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
41	28, 39, 40	sylancr 694	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
42	41	ralrimiva 2949	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43		iunfi 8137	. . . . . . . . 9 ⊢ (((◡(# ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → ∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
44	27, 42, 43	syl2anc 691	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
45		ficardom 8670	. . . . . . . 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		fvres 6117	. . . . . . 7 ⊢ ((card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω → ((# ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (#‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
48	46, 47	syl 17	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (#‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
49		hashcard 13008	. . . . . . 7 ⊢ (∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin → (#‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (#‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
50	44, 49	syl 17	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (#‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = (#‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
51		xp1st 7089	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) ∈ {𝑤})
52		elsni 4142	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ {𝑤} → (1st ‘𝑧) = 𝑤)
53	51, 52	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) = 𝑤)
54	53	rgen 2906	. . . . . . . . . 10 ⊢ ∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
55	54	rgenw 2908	. . . . . . . . 9 ⊢ ∀𝑤 ∈ (◡(# ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
56		invdisj 4571	. . . . . . . . 9 ⊢ (∀𝑤 ∈ (◡(# ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤 → Disj 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
57	55, 56	mp1i 13	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Disj 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
58	27, 41, 57	hashiun 14395	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (#‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ (◡(# ↾ ω) “ 𝑥)(#‘({𝑤} × 𝒫 𝑤)))
59		sneq 4135	. . . . . . . . . 10 ⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → {𝑤} = {(◡(# ↾ ω)‘𝑦)})
60		pweq 4111	. . . . . . . . . 10 ⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 (◡(# ↾ ω)‘𝑦))
61	59, 60	xpeq12d 5064	. . . . . . . . 9 ⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦)))
62	61	fveq2d 6107	. . . . . . . 8 ⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → (#‘({𝑤} × 𝒫 𝑤)) = (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))))
63		inss2 3796	. . . . . . . . 9 ⊢ (𝒫 ℕ0 ∩ Fin) ⊆ Fin
64	63	sseli 3564	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
65		f1of1 6049	. . . . . . . . . 10 ⊢ (◡(# ↾ ω):ℕ0–1-1-onto→ω → ◡(# ↾ ω):ℕ0–1-1→ω)
66	13, 65	ax-mp 5	. . . . . . . . 9 ⊢ ◡(# ↾ ω):ℕ0–1-1→ω
67		inss1 3795	. . . . . . . . . . 11 ⊢ (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0
68	67	sseli 3564	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
69	68	elpwid 4118	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
70		f1ores 6064	. . . . . . . . 9 ⊢ ((◡(# ↾ ω):ℕ0–1-1→ω ∧ 𝑥 ⊆ ℕ0) → (◡(# ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(# ↾ ω) “ 𝑥))
71	66, 69, 70	sylancr 694	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(# ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(# ↾ ω) “ 𝑥))
72		fvres 6117	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → ((◡(# ↾ ω) ↾ 𝑥)‘𝑦) = (◡(# ↾ ω)‘𝑦))
73	72	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((◡(# ↾ ω) ↾ 𝑥)‘𝑦) = (◡(# ↾ ω)‘𝑦))
74		hashcl 13009	. . . . . . . . 9 ⊢ (({𝑤} × 𝒫 𝑤) ∈ Fin → (#‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
75		nn0cn 11179	. . . . . . . . 9 ⊢ ((#‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0 → (#‘({𝑤} × 𝒫 𝑤)) ∈ ℂ)
76	41, 74, 75	3syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → (#‘({𝑤} × 𝒫 𝑤)) ∈ ℂ)
77	62, 64, 71, 73, 76	fsumf1o 14301	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ (◡(# ↾ ω) “ 𝑥)(#‘({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))))
78		snfi 7923	. . . . . . . . . 10 ⊢ {(◡(# ↾ ω)‘𝑦)} ∈ Fin
79	69	sselda 3568	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℕ0)
80		f1of 6050	. . . . . . . . . . . . . . 15 ⊢ (◡(# ↾ ω):ℕ0–1-1-onto→ω → ◡(# ↾ ω):ℕ0⟶ω)
81	13, 80	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ◡(# ↾ ω):ℕ0⟶ω
82	81	ffvelrni 6266	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0 → (◡(# ↾ ω)‘𝑦) ∈ ω)
83	79, 82	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (◡(# ↾ ω)‘𝑦) ∈ ω)
84	34, 83	sseldi 3566	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (◡(# ↾ ω)‘𝑦) ∈ Fin)
85		pwfi 8144	. . . . . . . . . . 11 ⊢ ((◡(# ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 (◡(# ↾ ω)‘𝑦) ∈ Fin)
86	84, 85	sylib 207	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝒫 (◡(# ↾ ω)‘𝑦) ∈ Fin)
87		hashxp 13081	. . . . . . . . . 10 ⊢ (({(◡(# ↾ ω)‘𝑦)} ∈ Fin ∧ 𝒫 (◡(# ↾ ω)‘𝑦) ∈ Fin) → (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = ((#‘{(◡(# ↾ ω)‘𝑦)}) · (#‘𝒫 (◡(# ↾ ω)‘𝑦))))
88	78, 86, 87	sylancr 694	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = ((#‘{(◡(# ↾ ω)‘𝑦)}) · (#‘𝒫 (◡(# ↾ ω)‘𝑦))))
89		hashsng 13020	. . . . . . . . . . 11 ⊢ ((◡(# ↾ ω)‘𝑦) ∈ ω → (#‘{(◡(# ↾ ω)‘𝑦)}) = 1)
90	83, 89	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘{(◡(# ↾ ω)‘𝑦)}) = 1)
91		hashpw 13083	. . . . . . . . . . . 12 ⊢ ((◡(# ↾ ω)‘𝑦) ∈ Fin → (#‘𝒫 (◡(# ↾ ω)‘𝑦)) = (2↑(#‘(◡(# ↾ ω)‘𝑦))))
92	84, 91	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘𝒫 (◡(# ↾ ω)‘𝑦)) = (2↑(#‘(◡(# ↾ ω)‘𝑦))))
93		fvres 6117	. . . . . . . . . . . . . 14 ⊢ ((◡(# ↾ ω)‘𝑦) ∈ ω → ((# ↾ ω)‘(◡(# ↾ ω)‘𝑦)) = (#‘(◡(# ↾ ω)‘𝑦)))
94	83, 93	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((# ↾ ω)‘(◡(# ↾ ω)‘𝑦)) = (#‘(◡(# ↾ ω)‘𝑦)))
95		f1ocnvfv2 6433	. . . . . . . . . . . . . 14 ⊢ (((# ↾ ω):ω–1-1-onto→ℕ0 ∧ 𝑦 ∈ ℕ0) → ((# ↾ ω)‘(◡(# ↾ ω)‘𝑦)) = 𝑦)
96	2, 79, 95	sylancr 694	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((# ↾ ω)‘(◡(# ↾ ω)‘𝑦)) = 𝑦)
97	94, 96	eqtr3d 2646	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘(◡(# ↾ ω)‘𝑦)) = 𝑦)
98	97	oveq2d 6565	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑(#‘(◡(# ↾ ω)‘𝑦))) = (2↑𝑦))
99	92, 98	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘𝒫 (◡(# ↾ ω)‘𝑦)) = (2↑𝑦))
100	90, 99	oveq12d 6567	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((#‘{(◡(# ↾ ω)‘𝑦)}) · (#‘𝒫 (◡(# ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
101		2cn 10968	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
102		expcl 12740	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
103	101, 79, 102	sylancr 694	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑𝑦) ∈ ℂ)
104	103	mulid2d 9937	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
105	88, 100, 104	3eqtrd 2648	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = (2↑𝑦))
106	105	sumeq2dv 14281	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦 ∈ 𝑥 (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
107	58, 77, 106	3eqtrd 2648	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (#‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
108	48, 50, 107	3eqtrd 2648	. . . . 5 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
109	108	mpteq2ia 4668	. . . 4 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (2↑𝑦))
110	46	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
111	26	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (◡(# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
112		eqidd 2611	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)))
113		eqidd 2611	. . . . . . 7 ⊢ (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))))
114		iuneq1 4470	. . . . . . . 8 ⊢ (𝑧 = (◡(# ↾ ω) “ 𝑥) → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
115	114	fveq2d 6107	. . . . . . 7 ⊢ (𝑧 = (◡(# ↾ ω) “ 𝑥) → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
116	111, 112, 113, 115	fmptco 6303	. . . . . 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))))
117		f1of 6050	. . . . . . . 8 ⊢ ((# ↾ ω):ω–1-1-onto→ℕ0 → (# ↾ ω):ω⟶ℕ0)
118	2, 117	mp1i 13	. . . . . . 7 ⊢ (⊤ → (# ↾ ω):ω⟶ℕ0)
119	118	feqmptd 6159	. . . . . 6 ⊢ (⊤ → (# ↾ ω) = (𝑦 ∈ ω ↦ ((# ↾ ω)‘𝑦)))
120		fveq2 6103	. . . . . 6 ⊢ (𝑦 = (card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) → ((# ↾ ω)‘𝑦) = ((# ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))))
121	110, 116, 119, 120	fmptco 6303	. . . . 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)))))
122	121	trud 1484	. . . 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))))
123		ackbijnn.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (2↑𝑦))
124	109, 122, 123	3eqtr4i 2642	. . 3 ⊢ ((# ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)))) = 𝐹
125		f1oeq1 6040	. . 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))
126	124, 125	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)
127	19, 126	mpbi 219	1 ⊢ 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ ciun 4455  Disj wdisj 4553   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037  dom cdm 5038   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Oncon0 5640  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ωcom 6957  1^st c1st 7057  Fincfn 7841  cardccrd 8644  ℂcc 9813  1c1 9816   · cmul 9820  2c2 10947  ℕ₀cn0 11169  ↑cexp 12722  #chash 12979  Σcsu 14264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265

This theorem is referenced by:  bitsinv2  15003