Theorem eulerpart 34948

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 2760	. . 3 ⊢ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
5		oveq2 7416	. . . 4 ⊢ (𝑎 = 𝑥 → ((2↑𝑏) · 𝑎) = ((2↑𝑏) · 𝑥))
6		oveq2 7416	. . . . 5 ⊢ (𝑏 = 𝑦 → (2↑𝑏) = (2↑𝑦))
7	6	oveq1d 7423	. . . 4 ⊢ (𝑏 = 𝑦 → ((2↑𝑏) · 𝑥) = ((2↑𝑦) · 𝑥))
8	5, 7	cbvmpov 7503	. . 3 ⊢ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) = (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
9		oveq1 7415	. . . . . 6 ⊢ (𝑟 = 𝑚 → (𝑟 supp ∅) = (𝑚 supp ∅))
10	9	eleq1d 2845	. . . . 5 ⊢ (𝑟 = 𝑚 → ((𝑟 supp ∅) ∈ Fin ↔ (𝑚 supp ∅) ∈ Fin))
11	10	cbvrabv 3422	. . . 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 2769	. . 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 6872	. . . . . . . 8 ⊢ (𝑡 = 𝑟 → (𝑡‘𝑎) = (𝑟‘𝑎))
14	13	eleq2d 2846	. . . . . . 7 ⊢ (𝑡 = 𝑟 → (𝑏 ∈ (𝑡‘𝑎) ↔ 𝑏 ∈ (𝑟‘𝑎)))
15	14	anbi2d 642	. . . . . 6 ⊢ (𝑡 = 𝑟 → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎)) ↔ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))))
16	15	opabbidv 5170	. . . . 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 5208	. . . 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 7415	. . . . . . . 8 ⊢ (𝑚 = 𝑠 → (𝑚 supp ∅) = (𝑠 supp ∅))
19	18	eleq1d 2845	. . . . . . 7 ⊢ (𝑚 = 𝑠 → ((𝑚 supp ∅) ∈ Fin ↔ (𝑠 supp ∅) ∈ Fin))
20	19	cbvrabv 3422	. . . . . 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 2769	. . . . 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 488	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 𝑎 = 𝑥)
23	22	eleq1d 2845	. . . . . . 7 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ↔ 𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
24		simpr 490	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 𝑏 = 𝑦)
25	22	fveq2d 6877	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑟‘𝑎) = (𝑟‘𝑥))
26	24, 25	eleq12d 2854	. . . . . . 7 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑏 ∈ (𝑟‘𝑎) ↔ 𝑦 ∈ (𝑟‘𝑥)))
27	23, 26	anbi12d 644	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎)) ↔ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))))
28	27	cbvopabv 5177	. . . . 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 5201	. . . 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 2783	. . 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 5847	. . . . . 6 ⊢ (ℎ = 𝑓 → ◡ℎ = ◡𝑓)
32	31	imaeq1d 6049	. . . . 5 ⊢ (ℎ = 𝑓 → (◡ℎ “ ℕ) = (◡𝑓 “ ℕ))
33	32	eleq1d 2845	. . . 4 ⊢ (ℎ = 𝑓 → ((◡ℎ “ ℕ) ∈ Fin ↔ (◡𝑓 “ ℕ) ∈ Fin))
34	33	cbvabv 2830	. . 3 ⊢ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin} = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
35	32	sseq1d 3961	. . . 4 ⊢ (ℎ = 𝑓 → ((◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ↔ (◡𝑓 “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
36	35	cbvrabv 3422	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}}
37		reseq1 5960	. . . . . . . . 9 ⊢ (𝑜 = 𝑞 → (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) = (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
38	37	coeq2d 5836	. . . . . . . 8 ⊢ (𝑜 = 𝑞 → (bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})) = (bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))
39	38	fveq2d 6877	. . . . . . 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 6050	. . . . . 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 6877	. . . . 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 5208	. . . 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 2769	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) = (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎))
44	43	imaeq1i 6047	. . . . . . 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 2760	. . . . . . . . . . 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 5201	. . . . . . . . . 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 6872	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑡 → (𝑟‘𝑎) = (𝑡‘𝑎))
48	47	eleq2d 2846	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑡 → (𝑏 ∈ (𝑟‘𝑎) ↔ 𝑏 ∈ (𝑡‘𝑎)))
49	48	anbi2d 642	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑡 → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎)) ↔ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))))
50	49	opabbidv 5170	. . . . . . . . . . 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 5208	. . . . . . . . . 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 2789	. . . . . . . . 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 6874	. . . . . . . 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 6048	. . . . . . 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 2783	. . . . . 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 6876	. . . . 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 5200	. . . 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 2783	. . 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 2760	. . 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 34947	. 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 7441	. . . . . . 7 ⊢ (ℕ0 ↑m ℕ) ∈ V
62	61	rabex 5299	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∈ V
63	62	inex1 5276	. . . . 5 ⊢ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ∈ V
64	63	mptex 7217	. . . 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 6016	. . 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 6800	. . 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 3560	. 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 8961	. . 3 ⊢ (𝑂 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝑂–1-1-onto→𝐷)
69		hasheni 14459	. . 3 ⊢ (𝑂 ≈ 𝐷 → (♯‘𝑂) = (♯‘𝐷))
70	68, 69	sylbir 238	. 2 ⊢ (∃𝑔 𝑔:𝑂–1-1-onto→𝐷 → (♯‘𝑂) = (♯‘𝐷))
71	60, 67, 70	mp2b 10	1 ⊢ (♯‘𝑂) = (♯‘𝐷)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∀wral 3076  {crab 3412   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  𝒫 cpw 4556   class class class wbr 5102  {copab 5166   ↦ cmpt 5185  ^◡ccnv 5646   ↾ cres 5649   “ cima 5650   ∘ ccom 5651  –1-1-onto→wf1o 6526  ‘cfv 6527  (class class class)co 7408   ∈ cmpo 7410   supp csupp 8155   ↑_m cmap 8825   ≈ cen 8948  Fincfn 8951  1c1 11172   · cmul 11176   ≤ cle 11315  𝟭cind 12289  ℕcn 12304  2c2 12366  ℕ₀cn0 12575  ↑cexp 14172  ♯chash 14441  Σcsu 15820   ∥ cdvds 16389  bitscbits 16556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248  ax-pre-sup 11249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  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 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-fsupp 9332  df-sup 9412  df-inf 9413  df-oi 9482  df-dju 9953  df-card 9991  df-acn 9994  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-div 11943  df-ind 12290  df-nn 12305  df-2 12374  df-3 12375  df-n0 12576  df-xnn0 12649  df-z 12663  df-uz 12935  df-rp 13090  df-fz 13609  df-fzo 13757  df-fl 13900  df-mod 13978  df-seq 14113  df-exp 14173  df-hash 14442  df-cj 15233  df-re 15234  df-im 15235  df-sqrt 15369  df-abs 15370  df-clim 15622  df-sum 15821  df-dvds 16390  df-bits 16559

This theorem is used by: (None)