Theorem ackbijnn 14560

Description: Translate the Ackermann bijection ackbij1 9060 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 13167	. . . 4 ⊢ (# ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
2	1	hashgf1o 12770	. . 3 ⊢ (# ↾ ω):ω–1-1-onto→ℕ0
3		sneq 4187	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → {𝑤} = {𝑦})
4		pweq 4161	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
5	3, 4	xpeq12d 5140	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ({𝑤} × 𝒫 𝑤) = ({𝑦} × 𝒫 𝑦))
6	5	cbviunv 4559	. . . . . . . 8 ⊢ ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦)
7		iuneq1 4534	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ∪ 𝑦 ∈ 𝑧 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
8	6, 7	syl5eq 2668	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))
9	8	fveq2d 6195	. . . . . 6 ⊢ (𝑧 = 𝑥 → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
10	9	cbvmptv 4750	. . . . 5 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
11	10	ackbij1 9060	. . . 4 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))):(𝒫 ω ∩ Fin)–1-1-onto→ω
12		f1ocnv 6149	. . . . . 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 8270	. . . . 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 6159	. . . 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 708	. . 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 6159	. . 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 708	. 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 3834	. . . . . . . . . 10 ⊢ (𝒫 ω ∩ Fin) ⊆ Fin
21		f1of 6137	. . . . . . . . . . . . 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 2622	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥))
24	23	fmpt 6381	. . . . . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ ∀𝑥 ∈ (𝒫 ℕ0 ∩ Fin)(◡(# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin)
26	25	rspec 2931	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
27	20, 26	sseldi 3601	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(# ↾ ω) “ 𝑥) ∈ Fin)
28		snfi 8038	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
29		cnvimass 5485	. . . . . . . . . . . . . . 15 ⊢ (◡(# ↾ ω) “ 𝑥) ⊆ dom (# ↾ ω)
30		dmhashres 13129	. . . . . . . . . . . . . . 15 ⊢ dom (# ↾ ω) = ω
31	29, 30	sseqtri 3637	. . . . . . . . . . . . . 14 ⊢ (◡(# ↾ ω) “ 𝑥) ⊆ ω
32		onfin2 8152	. . . . . . . . . . . . . . 15 ⊢ ω = (On ∩ Fin)
33		inss2 3834	. . . . . . . . . . . . . . 15 ⊢ (On ∩ Fin) ⊆ Fin
34	32, 33	eqsstri 3635	. . . . . . . . . . . . . 14 ⊢ ω ⊆ Fin
35	31, 34	sstri 3612	. . . . . . . . . . . . 13 ⊢ (◡(# ↾ ω) “ 𝑥) ⊆ Fin
36		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → 𝑤 ∈ (◡(# ↾ ω) “ 𝑥))
37	35, 36	sseldi 3601	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → 𝑤 ∈ Fin)
38		pwfi 8261	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Fin ↔ 𝒫 𝑤 ∈ Fin)
39	37, 38	sylib 208	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → 𝒫 𝑤 ∈ Fin)
40		xpfi 8231	. . . . . . . . . . 11 ⊢ (({𝑤} ∈ Fin ∧ 𝒫 𝑤 ∈ Fin) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
41	28, 39, 40	sylancr 695	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)) → ({𝑤} × 𝒫 𝑤) ∈ Fin)
42	41	ralrimiva 2966	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∀𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
43		iunfi 8254	. . . . . . . . 9 ⊢ (((◡(# ↾ ω) “ 𝑥) ∈ Fin ∧ ∀𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin) → ∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
44	27, 42, 43	syl2anc 693	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤) ∈ Fin)
45		ficardom 8787	. . . . . . . 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 6207	. . . . . . 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 13146	. . . . . . 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 7198	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) ∈ {𝑤})
52		elsni 4194	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ {𝑤} → (1st ‘𝑧) = 𝑤)
53	51, 52	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑤} × 𝒫 𝑤) → (1st ‘𝑧) = 𝑤)
54	53	rgen 2922	. . . . . . . . . 10 ⊢ ∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
55	54	rgenw 2924	. . . . . . . . 9 ⊢ ∀𝑤 ∈ (◡(# ↾ ω) “ 𝑥)∀𝑧 ∈ ({𝑤} × 𝒫 𝑤)(1st ‘𝑧) = 𝑤
56		invdisj 4638	. . . . . . . . 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 14554	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (#‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑤 ∈ (◡(# ↾ ω) “ 𝑥)(#‘({𝑤} × 𝒫 𝑤)))
59		sneq 4187	. . . . . . . . . 10 ⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → {𝑤} = {(◡(# ↾ ω)‘𝑦)})
60		pweq 4161	. . . . . . . . . 10 ⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → 𝒫 𝑤 = 𝒫 (◡(# ↾ ω)‘𝑦))
61	59, 60	xpeq12d 5140	. . . . . . . . 9 ⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → ({𝑤} × 𝒫 𝑤) = ({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦)))
62	61	fveq2d 6195	. . . . . . . 8 ⊢ (𝑤 = (◡(# ↾ ω)‘𝑦) → (#‘({𝑤} × 𝒫 𝑤)) = (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))))
63		inss2 3834	. . . . . . . . 9 ⊢ (𝒫 ℕ0 ∩ Fin) ⊆ Fin
64	63	sseli 3599	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
65		f1of1 6136	. . . . . . . . . 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 3833	. . . . . . . . . . 11 ⊢ (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0
68	67	sseli 3599	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ 𝒫 ℕ0)
69	68	elpwid 4170	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
70		f1ores 6151	. . . . . . . . 9 ⊢ ((◡(# ↾ ω):ℕ0–1-1→ω ∧ 𝑥 ⊆ ℕ0) → (◡(# ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(# ↾ ω) “ 𝑥))
71	66, 69, 70	sylancr 695	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (◡(# ↾ ω) ↾ 𝑥):𝑥–1-1-onto→(◡(# ↾ ω) “ 𝑥))
72		fvres 6207	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑥 → ((◡(# ↾ ω) ↾ 𝑥)‘𝑦) = (◡(# ↾ ω)‘𝑦))
73	72	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((◡(# ↾ ω) ↾ 𝑥)‘𝑦) = (◡(# ↾ ω)‘𝑦))
74		hashcl 13147	. . . . . . . . 9 ⊢ (({𝑤} × 𝒫 𝑤) ∈ Fin → (#‘({𝑤} × 𝒫 𝑤)) ∈ ℕ0)
75		nn0cn 11302	. . . . . . . . 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 14454	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑤 ∈ (◡(# ↾ ω) “ 𝑥)(#‘({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))))
78		snfi 8038	. . . . . . . . . 10 ⊢ {(◡(# ↾ ω)‘𝑦)} ∈ Fin
79	69	sselda 3603	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℕ0)
80		f1of 6137	. . . . . . . . . . . . . . 15 ⊢ (◡(# ↾ ω):ℕ0–1-1-onto→ω → ◡(# ↾ ω):ℕ0⟶ω)
81	13, 80	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ◡(# ↾ ω):ℕ0⟶ω
82	81	ffvelrni 6358	. . . . . . . . . . . . 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 3601	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (◡(# ↾ ω)‘𝑦) ∈ Fin)
85		pwfi 8261	. . . . . . . . . . 11 ⊢ ((◡(# ↾ ω)‘𝑦) ∈ Fin ↔ 𝒫 (◡(# ↾ ω)‘𝑦) ∈ Fin)
86	84, 85	sylib 208	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝒫 (◡(# ↾ ω)‘𝑦) ∈ Fin)
87		hashxp 13221	. . . . . . . . . 10 ⊢ (({(◡(# ↾ ω)‘𝑦)} ∈ Fin ∧ 𝒫 (◡(# ↾ ω)‘𝑦) ∈ Fin) → (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = ((#‘{(◡(# ↾ ω)‘𝑦)}) · (#‘𝒫 (◡(# ↾ ω)‘𝑦))))
88	78, 86, 87	sylancr 695	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = ((#‘{(◡(# ↾ ω)‘𝑦)}) · (#‘𝒫 (◡(# ↾ ω)‘𝑦))))
89		hashsng 13159	. . . . . . . . . . 11 ⊢ ((◡(# ↾ ω)‘𝑦) ∈ ω → (#‘{(◡(# ↾ ω)‘𝑦)}) = 1)
90	83, 89	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘{(◡(# ↾ ω)‘𝑦)}) = 1)
91		hashpw 13223	. . . . . . . . . . . 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 6207	. . . . . . . . . . . . . 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 6533	. . . . . . . . . . . . . 14 ⊢ (((# ↾ ω):ω–1-1-onto→ℕ0 ∧ 𝑦 ∈ ℕ0) → ((# ↾ ω)‘(◡(# ↾ ω)‘𝑦)) = 𝑦)
96	2, 79, 95	sylancr 695	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((# ↾ ω)‘(◡(# ↾ ω)‘𝑦)) = 𝑦)
97	94, 96	eqtr3d 2658	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘(◡(# ↾ ω)‘𝑦)) = 𝑦)
98	97	oveq2d 6666	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑(#‘(◡(# ↾ ω)‘𝑦))) = (2↑𝑦))
99	92, 98	eqtrd 2656	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘𝒫 (◡(# ↾ ω)‘𝑦)) = (2↑𝑦))
100	90, 99	oveq12d 6668	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((#‘{(◡(# ↾ ω)‘𝑦)}) · (#‘𝒫 (◡(# ↾ ω)‘𝑦))) = (1 · (2↑𝑦)))
101		2cn 11091	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
102		expcl 12878	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℂ)
103	101, 79, 102	sylancr 695	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (2↑𝑦) ∈ ℂ)
104	103	mulid2d 10058	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (1 · (2↑𝑦)) = (2↑𝑦))
105	88, 100, 104	3eqtrd 2660	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = (2↑𝑦))
106	105	sumeq2dv 14433	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑦 ∈ 𝑥 (#‘({(◡(# ↾ ω)‘𝑦)} × 𝒫 (◡(# ↾ ω)‘𝑦))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
107	58, 77, 106	3eqtrd 2660	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (#‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
108	48, 50, 107	3eqtrd 2660	. . . . 5 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → ((# ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))) = Σ𝑦 ∈ 𝑥 (2↑𝑦))
109	108	mpteq2ia 4740	. . . 4 ⊢ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ ((# ↾ ω)‘(card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (2↑𝑦))
110	46	adantl 482	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)) ∈ ω)
111	26	adantl 482	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → (◡(# ↾ ω) “ 𝑥) ∈ (𝒫 ω ∩ Fin))
112		eqidd 2623	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)))
113		eqidd 2623	. . . . . . 7 ⊢ (⊤ → (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))))
114		iuneq1 4534	. . . . . . . 8 ⊢ (𝑧 = (◡(# ↾ ω) “ 𝑥) → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤))
115	114	fveq2d 6195	. . . . . . 7 ⊢ (𝑧 = (◡(# ↾ ω) “ 𝑥) → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑤 ∈ (◡(# ↾ ω) “ 𝑥)({𝑤} × 𝒫 𝑤)))
116	111, 112, 113, 115	fmptco 6396	. . . . . 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 6137	. . . . . . . 8 ⊢ ((# ↾ ω):ω–1-1-onto→ℕ0 → (# ↾ ω):ω⟶ℕ0)
118	2, 117	mp1i 13	. . . . . . 7 ⊢ (⊤ → (# ↾ ω):ω⟶ℕ0)
119	118	feqmptd 6249	. . . . . 6 ⊢ (⊤ → (# ↾ ω) = (𝑦 ∈ ω ↦ ((# ↾ ω)‘𝑦)))
120		fveq2 6191	. . . . . 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 6396	. . . . 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 1493	. . . 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 2654	. . 3 ⊢ ((# ↾ ω) ∘ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) ∘ (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ (◡(# ↾ ω) “ 𝑥)))) = 𝐹
125		f1oeq1 6127	. . 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 220	1 ⊢ 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  ∀wral 2912   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ∪ ciun 4520  Disj wdisj 4620   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   ↾ cres 5116   “ cima 5117   ∘ ccom 5118  Oncon0 5723  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  ωcom 7065  1^st c1st 7166  Fincfn 7955  cardccrd 8761  ℂcc 9934  1c1 9937   · cmul 9941  2c2 11070  ℕ₀cn0 11292  ↑cexp 12860  #chash 13117  Σcsu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  bitsinv2  15165