Theorem eulerpart 32249

Description: Euler's theorem on partitions, also known as a special case of Glaisher's theorem. Let 𝑃 be the set of all partitions of 𝑁, represented as multisets of positive integers, which is to say functions from ℕ to ℕ₀ where the value of the function represents the number of repetitions of an individual element, and the sum of all the elements with repetition equals 𝑁. Then the set 𝑂 of all partitions that only consist of odd numbers and the set 𝐷 of all partitions which have no repeated elements have the same cardinality. This is Metamath 100 proof #45. (Contributed by Thierry Arnoux, 14-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)

Hypotheses

Ref	Expression
eulerpart.p	⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
eulerpart.o	⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d	⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}

Assertion

Ref	Expression
eulerpart	⊢ (♯‘𝑂) = (♯‘𝐷)

Distinct variable groups:   𝑓,𝑔,𝑘,𝑛   𝐷,𝑔   𝑓,𝑁,𝑔,𝑘,𝑛   𝑔,𝑂,𝑛   𝑃,𝑔,𝑘,𝑛

Allowed substitution hints:   𝐷(𝑓,𝑘,𝑛)   𝑃(𝑓)   𝑂(𝑓,𝑘)

Proof of Theorem eulerpart

Dummy variables 𝑎 𝑏 ℎ 𝑚 𝑜 𝑞 𝑟 𝑠 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eulerpart.p	. . 3 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
2		eulerpart.o	. . 3 ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
3		eulerpart.d	. . 3 ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
4		eqid 2738	. . 3 ⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
5		oveq2 7263	. . . 4 ⊢ (𝑎 = 𝑥 → ((2↑𝑏) · 𝑎) = ((2↑𝑏) · 𝑥))
6		oveq2 7263	. . . . 5 ⊢ (𝑏 = 𝑦 → (2↑𝑏) = (2↑𝑦))
7	6	oveq1d 7270	. . . 4 ⊢ (𝑏 = 𝑦 → ((2↑𝑏) · 𝑥) = ((2↑𝑦) · 𝑥))
8	5, 7	cbvmpov 7348	. . 3 ⊢ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) = (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
9		oveq1 7262	. . . . . 6 ⊢ (𝑟 = 𝑚 → (𝑟 supp ∅) = (𝑚 supp ∅))
10	9	eleq1d 2823	. . . . 5 ⊢ (𝑟 = 𝑚 → ((𝑟 supp ∅) ∈ Fin ↔ (𝑚 supp ∅) ∈ Fin))
11	10	cbvrabv 3416	. . . 4 ⊢ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} = {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin}
12	11	eqcomi 2747	. . 3 ⊢ {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin} = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin}
13		fveq1 6755	. . . . . . . 8 ⊢ (𝑡 = 𝑟 → (𝑡‘𝑎) = (𝑟‘𝑎))
14	13	eleq2d 2824	. . . . . . 7 ⊢ (𝑡 = 𝑟 → (𝑏 ∈ (𝑡‘𝑎) ↔ 𝑏 ∈ (𝑟‘𝑎)))
15	14	anbi2d 628	. . . . . 6 ⊢ (𝑡 = 𝑟 → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎)) ↔ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))))
16	15	opabbidv 5136	. . . . 5 ⊢ (𝑡 = 𝑟 → {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))})
17	16	cbvmptv 5183	. . . 4 ⊢ (𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))}) = (𝑟 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))})
18		oveq1 7262	. . . . . . . 8 ⊢ (𝑚 = 𝑠 → (𝑚 supp ∅) = (𝑠 supp ∅))
19	18	eleq1d 2823	. . . . . . 7 ⊢ (𝑚 = 𝑠 → ((𝑚 supp ∅) ∈ Fin ↔ (𝑠 supp ∅) ∈ Fin))
20	19	cbvrabv 3416	. . . . . 6 ⊢ {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin} = {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin}
21	20	eqcomi 2747	. . . . 5 ⊢ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} = {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin}
22		simpl 482	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 𝑎 = 𝑥)
23	22	eleq1d 2823	. . . . . . 7 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ↔ 𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
24		simpr 484	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 𝑏 = 𝑦)
25	22	fveq2d 6760	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑟‘𝑎) = (𝑟‘𝑥))
26	24, 25	eleq12d 2833	. . . . . . 7 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑏 ∈ (𝑟‘𝑎) ↔ 𝑦 ∈ (𝑟‘𝑥)))
27	23, 26	anbi12d 630	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎)) ↔ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))))
28	27	cbvopabv 5143	. . . . 5 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))}
29	21, 28	mpteq12i 5176	. . . 4 ⊢ (𝑟 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))}) = (𝑟 ∈ {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})
30	17, 29	eqtri 2766	. . 3 ⊢ (𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))}) = (𝑟 ∈ {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})
31		cnveq 5771	. . . . . 6 ⊢ (ℎ = 𝑓 → ◡ℎ = ◡𝑓)
32	31	imaeq1d 5957	. . . . 5 ⊢ (ℎ = 𝑓 → (◡ℎ “ ℕ) = (◡𝑓 “ ℕ))
33	32	eleq1d 2823	. . . 4 ⊢ (ℎ = 𝑓 → ((◡ℎ “ ℕ) ∈ Fin ↔ (◡𝑓 “ ℕ) ∈ Fin))
34	33	cbvabv 2812	. . 3 ⊢ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin} = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
35	32	sseq1d 3948	. . . 4 ⊢ (ℎ = 𝑓 → ((◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ↔ (◡𝑓 “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
36	35	cbvrabv 3416	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}}
37		reseq1 5874	. . . . . . . . 9 ⊢ (𝑜 = 𝑞 → (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) = (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
38	37	coeq2d 5760	. . . . . . . 8 ⊢ (𝑜 = 𝑞 → (bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})) = (bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))
39	38	fveq2d 6760	. . . . . . 7 ⊢ (𝑜 = 𝑞 → ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))) = ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))
40	39	imaeq2d 5958	. . . . . 6 ⊢ (𝑜 = 𝑞 → ((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))) = ((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))
41	40	fveq2d 6760	. . . . 5 ⊢ (𝑜 = 𝑞 → ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))) = ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))))
42	41	cbvmptv 5183	. . . 4 ⊢ (𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) = (𝑞 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))))
43	8	eqcomi 2747	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) = (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎))
44	43	imaeq1i 5955	. . . . . . 7 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))) = ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))
45		eqid 2738	. . . . . . . . . . 11 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))}
46	11, 45	mpteq12i 5176	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))}) = (𝑟 ∈ {𝑚 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑚 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})
47		fveq1 6755	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑡 → (𝑟‘𝑎) = (𝑡‘𝑎))
48	47	eleq2d 2824	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑡 → (𝑏 ∈ (𝑟‘𝑎) ↔ 𝑏 ∈ (𝑡‘𝑎)))
49	48	anbi2d 628	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑡 → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎)) ↔ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))))
50	49	opabbidv 5136	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑡 → {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})
51	50	cbvmptv 5183	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))}) = (𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})
52	46, 29, 51	3eqtr2i 2772	. . . . . . . . 9 ⊢ (𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))}) = (𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})
53	52	fveq1i 6757	. . . . . . . 8 ⊢ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))) = ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))
54	53	imaeq2i 5956	. . . . . . 7 ⊢ ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))) = ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))
55	44, 54	eqtri 2766	. . . . . 6 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))) = ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))
56	55	fveq2i 6759	. . . . 5 ⊢ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))) = ((𝟭‘ℕ)‘((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))
57	56	mpteq2i 5175	. . . 4 ⊢ (𝑞 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) = (𝑞 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))))
58	42, 57	eqtri 2766	. . 3 ⊢ (𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) = (𝑞 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) “ ((𝑡 ∈ {𝑠 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑠 supp ∅) ∈ Fin} ↦ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))})‘(bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))))))
59		eqid 2738	. . 3 ⊢ (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))
60	1, 2, 3, 4, 8, 12, 30, 34, 36, 58, 59	eulerpartlemn 32248	. 2 ⊢ ((𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ↾ 𝑂):𝑂–1-1-onto→𝐷
61		ovex 7288	. . . . . . 7 ⊢ (ℕ0 ↑m ℕ) ∈ V
62	61	rabex 5251	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∈ V
63	62	inex1 5236	. . . . 5 ⊢ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ∈ V
64	63	mptex 7081	. . . 4 ⊢ (𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ∈ V
65	64	resex 5928	. . 3 ⊢ ((𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ↾ 𝑂) ∈ V
66		f1oeq1 6688	. . 3 ⊢ (𝑔 = ((𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ↾ 𝑂) → (𝑔:𝑂–1-1-onto→𝐷 ↔ ((𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ↾ 𝑂):𝑂–1-1-onto→𝐷))
67	65, 66	spcev 3535	. 2 ⊢ (((𝑜 ∈ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ↦ ((𝟭‘ℕ)‘((𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) “ ((𝑟 ∈ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) ∣ (𝑟 supp ∅) ∈ Fin} ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))})‘(bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))))) ↾ 𝑂):𝑂–1-1-onto→𝐷 → ∃𝑔 𝑔:𝑂–1-1-onto→𝐷)
68		bren 8701	. . 3 ⊢ (𝑂 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝑂–1-1-onto→𝐷)
69		hasheni 13990	. . 3 ⊢ (𝑂 ≈ 𝐷 → (♯‘𝑂) = (♯‘𝐷))
70	68, 69	sylbir 234	. 2 ⊢ (∃𝑔 𝑔:𝑂–1-1-onto→𝐷 → (♯‘𝑂) = (♯‘𝐷))
71	60, 67, 70	mp2b 10	1 ⊢ (♯‘𝑂) = (♯‘𝐷)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∀wral 3063  {crab 3067   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530   class class class wbr 5070  {copab 5132   ↦ cmpt 5153  ^◡ccnv 5579   ↾ cres 5582   “ cima 5583   ∘ ccom 5584  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   supp csupp 7948   ↑_m cmap 8573   ≈ cen 8688  Fincfn 8691  1c1 10803   · cmul 10807   ≤ cle 10941  ℕcn 11903  2c2 11958  ℕ₀cn0 12163  ↑cexp 13710  ♯chash 13972  Σcsu 15325   ∥ cdvds 15891  bitscbits 16054  𝟭cind 31878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-ac2 10150  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-acn 9631  df-ac 9803  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-rp 12660  df-fz 13169  df-fzo 13312  df-fl 13440  df-mod 13518  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-dvds 15892  df-bits 16057  df-ind 31879

This theorem is referenced by: (None)