Theorem eulerpart 32349

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 7283	. . . 4 ⊢ (𝑎 = 𝑥 → ((2↑𝑏) · 𝑎) = ((2↑𝑏) · 𝑥))
6		oveq2 7283	. . . . 5 ⊢ (𝑏 = 𝑦 → (2↑𝑏) = (2↑𝑦))
7	6	oveq1d 7290	. . . 4 ⊢ (𝑏 = 𝑦 → ((2↑𝑏) · 𝑥) = ((2↑𝑦) · 𝑥))
8	5, 7	cbvmpov 7370	. . 3 ⊢ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑏 ∈ ℕ0 ↦ ((2↑𝑏) · 𝑎)) = (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
9		oveq1 7282	. . . . . 6 ⊢ (𝑟 = 𝑚 → (𝑟 supp ∅) = (𝑚 supp ∅))
10	9	eleq1d 2823	. . . . 5 ⊢ (𝑟 = 𝑚 → ((𝑟 supp ∅) ∈ Fin ↔ (𝑚 supp ∅) ∈ Fin))
11	10	cbvrabv 3426	. . . 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 6773	. . . . . . . 8 ⊢ (𝑡 = 𝑟 → (𝑡‘𝑎) = (𝑟‘𝑎))
14	13	eleq2d 2824	. . . . . . 7 ⊢ (𝑡 = 𝑟 → (𝑏 ∈ (𝑡‘𝑎) ↔ 𝑏 ∈ (𝑟‘𝑎)))
15	14	anbi2d 629	. . . . . 6 ⊢ (𝑡 = 𝑟 → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎)) ↔ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎))))
16	15	opabbidv 5140	. . . . 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 5187	. . . 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 7282	. . . . . . . 8 ⊢ (𝑚 = 𝑠 → (𝑚 supp ∅) = (𝑠 supp ∅))
19	18	eleq1d 2823	. . . . . . 7 ⊢ (𝑚 = 𝑠 → ((𝑚 supp ∅) ∈ Fin ↔ (𝑠 supp ∅) ∈ Fin))
20	19	cbvrabv 3426	. . . . . 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 483	. . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 𝑏 = 𝑦)
25	22	fveq2d 6778	. . . . . . . 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 631	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎)) ↔ (𝑥 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑦 ∈ (𝑟‘𝑥))))
28	27	cbvopabv 5147	. . . . 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 5180	. . . 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 5782	. . . . . 6 ⊢ (ℎ = 𝑓 → ◡ℎ = ◡𝑓)
32	31	imaeq1d 5968	. . . . 5 ⊢ (ℎ = 𝑓 → (◡ℎ “ ℕ) = (◡𝑓 “ ℕ))
33	32	eleq1d 2823	. . . 4 ⊢ (ℎ = 𝑓 → ((◡ℎ “ ℕ) ∈ Fin ↔ (◡𝑓 “ ℕ) ∈ Fin))
34	33	cbvabv 2811	. . 3 ⊢ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin} = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
35	32	sseq1d 3952	. . . 4 ⊢ (ℎ = 𝑓 → ((◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ↔ (◡𝑓 “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
36	35	cbvrabv 3426	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}}
37		reseq1 5885	. . . . . . . . 9 ⊢ (𝑜 = 𝑞 → (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}) = (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}))
38	37	coeq2d 5771	. . . . . . . 8 ⊢ (𝑜 = 𝑞 → (bits ∘ (𝑜 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})) = (bits ∘ (𝑞 ↾ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧})))
39	38	fveq2d 6778	. . . . . . 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 5969	. . . . . 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 6778	. . . . 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 5187	. . . 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 5966	. . . . . . 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 5180	. . . . . . . . . 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 6773	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑡 → (𝑟‘𝑎) = (𝑡‘𝑎))
48	47	eleq2d 2824	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑡 → (𝑏 ∈ (𝑟‘𝑎) ↔ 𝑏 ∈ (𝑡‘𝑎)))
49	48	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑡 → ((𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑟‘𝑎)) ↔ (𝑎 ∈ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} ∧ 𝑏 ∈ (𝑡‘𝑎))))
50	49	opabbidv 5140	. . . . . . . . . . 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 5187	. . . . . . . . . 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 6775	. . . . . . . 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 5967	. . . . . . 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 6777	. . . . 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 5179	. . . 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 32348	. 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 7308	. . . . . . 7 ⊢ (ℕ0 ↑m ℕ) ∈ V
62	61	rabex 5256	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∈ V
63	62	inex1 5241	. . . . 5 ⊢ ({ℎ ∈ (ℕ0 ↑m ℕ) ∣ (◡ℎ “ ℕ) ⊆ {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}} ∩ {ℎ ∣ (◡ℎ “ ℕ) ∈ Fin}) ∈ V
64	63	mptex 7099	. . . 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 5939	. . 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 6704	. . 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 3545	. 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 8743	. . 3 ⊢ (𝑂 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝑂–1-1-onto→𝐷)
69		hasheni 14062	. . 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 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∀wral 3064  {crab 3068   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533   class class class wbr 5074  {copab 5136   ↦ cmpt 5157  ^◡ccnv 5588   ↾ cres 5591   “ cima 5592   ∘ ccom 5593  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   supp csupp 7977   ↑_m cmap 8615   ≈ cen 8730  Fincfn 8733  1c1 10872   · cmul 10876   ≤ cle 11010  ℕcn 11973  2c2 12028  ℕ₀cn0 12233  ↑cexp 13782  ♯chash 14044  Σcsu 15397   ∥ cdvds 15963  bitscbits 16126  𝟭cind 31978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-ac2 10219  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-acn 9700  df-ac 9872  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398  df-dvds 15964  df-bits 16129  df-ind 31979

This theorem is referenced by: (None)