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Theorem bitsf1 16387

Description: The bits function is an injection from ℤ to 𝒫 ℕ₀. It is obviously not a bijection (by Cantor's theorem canth2 9130), and in fact its range is the set of finite and cofinite subsets of ℕ₀. (Contributed by Mario Carneiro, 22-Sep-2016.)

**Assertion**
Ref	Expression
bitsf1	⊢ bits:ℤ–1-1→𝒫 ℕ₀

Proof of Theorem bitsf1 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bitsf 16368	. 2 ⊢ bits:ℤ⟶𝒫 ℕ₀
2		simpl 484	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℤ)
3	2	zcnd 12667	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℂ)
4	3	adantr 482	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℂ)
5		simpr 486	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
6	5	zcnd 12667	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℂ)
7	6	adantr 482	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑦 ∈ ℂ)
8	4	negcld 11558	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 ∈ ℂ)
9	7	negcld 11558	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℂ)
10		1cnd 11209	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 1 ∈ ℂ)
11		simprr 772	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
12	11	difeq2d 4123	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ₀ ∖ (bits‘𝑥)) = (ℕ₀ ∖ (bits‘𝑦)))
13		bitscmp 16379	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℤ → (ℕ₀ ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
14	13	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ₀ ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
15		bitscmp 16379	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℤ → (ℕ₀ ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
16	15	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ₀ ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
17	12, 14, 16	3eqtr3d 2781	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘(-𝑥 − 1)) = (bits‘(-𝑦 − 1)))
18		nnm1nn0 12513	. . . . . . . . . . 11 ⊢ (-𝑥 ∈ ℕ → (-𝑥 − 1) ∈ ℕ₀)
19	18	ad2antrl 727	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) ∈ ℕ₀)
20	19	fvresd 6912	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ₀)‘(-𝑥 − 1)) = (bits‘(-𝑥 − 1)))
21		ominf 9258	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ω ∈ Fin
22		nn0ennn 13944	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ₀ ≈ ℕ
23		nnenom 13945	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ ≈ ω
24	22, 23	entr2i 9005	. . . . . . . . . . . . . . . . . 18 ⊢ ω ≈ ℕ₀
25		enfii 9189	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℕ₀ ∈ Fin ∧ ω ≈ ℕ₀) → ω ∈ Fin)
26	24, 25	mpan2 690	. . . . . . . . . . . . . . . . 17 ⊢ (ℕ₀ ∈ Fin → ω ∈ Fin)
27	21, 26	mto 196	. . . . . . . . . . . . . . . 16 ⊢ ¬ ℕ₀ ∈ Fin
28		difinf 9316	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ℕ₀ ∈ Fin ∧ (bits‘𝑥) ∈ Fin) → ¬ (ℕ₀ ∖ (bits‘𝑥)) ∈ Fin)
29	27, 28	mpan 689	. . . . . . . . . . . . . . 15 ⊢ ((bits‘𝑥) ∈ Fin → ¬ (ℕ₀ ∖ (bits‘𝑥)) ∈ Fin)
30		bitsfi 16378	. . . . . . . . . . . . . . . . 17 ⊢ ((-𝑥 − 1) ∈ ℕ₀ → (bits‘(-𝑥 − 1)) ∈ Fin)
31	19, 30	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘(-𝑥 − 1)) ∈ Fin)
32	14, 31	eqeltrd 2834	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ₀ ∖ (bits‘𝑥)) ∈ Fin)
33	29, 32	nsyl3 138	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑥) ∈ Fin)
34	11, 33	eqneltrrd 2855	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑦) ∈ Fin)
35		bitsfi 16378	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ₀ → (bits‘𝑦) ∈ Fin)
36	34, 35	nsyl 140	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ 𝑦 ∈ ℕ₀)
37	5	znegcld 12668	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
38		elznn 12574	. . . . . . . . . . . . . . . . 17 ⊢ (-𝑦 ∈ ℤ ↔ (-𝑦 ∈ ℝ ∧ (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ₀)))
39	38	simprbi 498	. . . . . . . . . . . . . . . 16 ⊢ (-𝑦 ∈ ℤ → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ₀))
40	37, 39	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ₀))
41	6	negnegd 11562	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑦 = 𝑦)
42	41	eleq1d 2819	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑦 ∈ ℕ₀ ↔ 𝑦 ∈ ℕ₀))
43	42	orbi2d 915	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ₀) ↔ (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ₀)))
44	40, 43	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ₀))
45	44	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ₀))
46	45	ord 863	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (¬ -𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀))
47	36, 46	mt3d 148	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℕ)
48		nnm1nn0 12513	. . . . . . . . . . 11 ⊢ (-𝑦 ∈ ℕ → (-𝑦 − 1) ∈ ℕ₀)
49	47, 48	syl 17	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 − 1) ∈ ℕ₀)
50	49	fvresd 6912	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ₀)‘(-𝑦 − 1)) = (bits‘(-𝑦 − 1)))
51	17, 20, 50	3eqtr4d 2783	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ₀)‘(-𝑥 − 1)) = ((bits ↾ ℕ₀)‘(-𝑦 − 1)))
52		bitsf1o 16386	. . . . . . . . . . 11 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin)
53		f1of1 6833	. . . . . . . . . . 11 ⊢ ((bits ↾ ℕ₀):ℕ₀–1-1-onto→(𝒫 ℕ₀ ∩ Fin) → (bits ↾ ℕ₀):ℕ₀–1-1→(𝒫 ℕ₀ ∩ Fin))
54	52, 53	ax-mp 5	. . . . . . . . . 10 ⊢ (bits ↾ ℕ₀):ℕ₀–1-1→(𝒫 ℕ₀ ∩ Fin)
55		f1fveq 7261	. . . . . . . . . 10 ⊢ (((bits ↾ ℕ₀):ℕ₀–1-1→(𝒫 ℕ₀ ∩ Fin) ∧ ((-𝑥 − 1) ∈ ℕ₀ ∧ (-𝑦 − 1) ∈ ℕ₀)) → (((bits ↾ ℕ₀)‘(-𝑥 − 1)) = ((bits ↾ ℕ₀)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
56	54, 55	mpan 689	. . . . . . . . 9 ⊢ (((-𝑥 − 1) ∈ ℕ₀ ∧ (-𝑦 − 1) ∈ ℕ₀) → (((bits ↾ ℕ₀)‘(-𝑥 − 1)) = ((bits ↾ ℕ₀)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
57	19, 49, 56	syl2anc 585	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (((bits ↾ ℕ₀)‘(-𝑥 − 1)) = ((bits ↾ ℕ₀)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
58	51, 57	mpbid 231	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) = (-𝑦 − 1))
59	8, 9, 10, 58	subcan2d 11613	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 = -𝑦)
60	4, 7, 59	neg11d 11583	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
61	60	expr 458	. . . 4 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ -𝑥 ∈ ℕ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
62	3	negnegd 11562	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑥 = 𝑥)
63	62	eleq1d 2819	. . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑥 ∈ ℕ₀ ↔ 𝑥 ∈ ℕ₀))
64	63	biimpa 478	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ₀) → 𝑥 ∈ ℕ₀)
65		simprr 772	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
66		fvres 6911	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ₀ → ((bits ↾ ℕ₀)‘𝑥) = (bits‘𝑥))
67	66	ad2antrl 727	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ₀)‘𝑥) = (bits‘𝑥))
68	15	ad2antlr 726	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ₀ ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
69		bitsfi 16378	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ₀ → (bits‘𝑥) ∈ Fin)
70	69	ad2antrl 727	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) ∈ Fin)
71	65, 70	eqeltrrd 2835	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑦) ∈ Fin)
72		difinf 9316	. . . . . . . . . . . . . 14 ⊢ ((¬ ℕ₀ ∈ Fin ∧ (bits‘𝑦) ∈ Fin) → ¬ (ℕ₀ ∖ (bits‘𝑦)) ∈ Fin)
73	27, 71, 72	sylancr 588	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (ℕ₀ ∖ (bits‘𝑦)) ∈ Fin)
74	68, 73	eqneltrrd 2855	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘(-𝑦 − 1)) ∈ Fin)
75		bitsfi 16378	. . . . . . . . . . . 12 ⊢ ((-𝑦 − 1) ∈ ℕ₀ → (bits‘(-𝑦 − 1)) ∈ Fin)
76	74, 75	nsyl 140	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (-𝑦 − 1) ∈ ℕ₀)
77	76, 48	nsyl 140	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ -𝑦 ∈ ℕ)
78	44	adantr 482	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ₀))
79	78	ord 863	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → (¬ -𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀))
80	77, 79	mpd 15	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑦 ∈ ℕ₀)
81	80	fvresd 6912	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ₀)‘𝑦) = (bits‘𝑦))
82	65, 67, 81	3eqtr4d 2783	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ₀)‘𝑥) = ((bits ↾ ℕ₀)‘𝑦))
83		simprl 770	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℕ₀)
84		f1fveq 7261	. . . . . . . . 9 ⊢ (((bits ↾ ℕ₀):ℕ₀–1-1→(𝒫 ℕ₀ ∩ Fin) ∧ (𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀)) → (((bits ↾ ℕ₀)‘𝑥) = ((bits ↾ ℕ₀)‘𝑦) ↔ 𝑥 = 𝑦))
85	54, 84	mpan 689	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀) → (((bits ↾ ℕ₀)‘𝑥) = ((bits ↾ ℕ₀)‘𝑦) ↔ 𝑥 = 𝑦))
86	83, 80, 85	syl2anc 585	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → (((bits ↾ ℕ₀)‘𝑥) = ((bits ↾ ℕ₀)‘𝑦) ↔ 𝑥 = 𝑦))
87	82, 86	mpbid 231	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ₀ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
88	87	expr 458	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ 𝑥 ∈ ℕ₀) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
89	64, 88	syldan 592	. . . 4 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ₀) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
90	2	znegcld 12668	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑥 ∈ ℤ)
91		elznn 12574	. . . . . 6 ⊢ (-𝑥 ∈ ℤ ↔ (-𝑥 ∈ ℝ ∧ (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ₀)))
92	91	simprbi 498	. . . . 5 ⊢ (-𝑥 ∈ ℤ → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ₀))
93	90, 92	syl 17	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ₀))
94	61, 89, 93	mpjaodan 958	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
95	94	rgen2 3198	. 2 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦)
96		dff13 7254	. 2 ⊢ (bits:ℤ–1-1→𝒫 ℕ₀ ↔ (bits:ℤ⟶𝒫 ℕ₀ ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦)))
97	1, 95, 96	mpbir2an 710	1 ⊢ bits:ℤ–1-1→𝒫 ℕ₀

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∖ cdif 3946 ∩ cin 3948 𝒫 cpw 4603 class class class wbr 5149 ↾ cres 5679 ⟶wf 6540 –1-1→wf1 6541 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7409 ωcom 7855 ≈ cen 8936 Fincfn 8939 ℂcc 11108 ℝcr 11109 1c1 11111 − cmin 11444 -cneg 11445 ℕcn 12212 ℕ₀cn0 12472 ℤcz 12558 bitscbits 16360

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-dvds 16198 df-bits 16363

This theorem is referenced by: bitsuz 16415 eulerpartlemmf 33405

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