Theorem bitsf1 15541

Description: The bits function is an injection from ℤ to 𝒫 ℕ₀. It is obviously not a bijection (by Cantor's theorem canth2 8382), 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 15522	. 2 ⊢ bits:ℤ⟶𝒫 ℕ0
2		simpl 476	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℤ)
3	2	zcnd 11811	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℂ)
4	3	adantr 474	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℂ)
5		simpr 479	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
6	5	zcnd 11811	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℂ)
7	6	adantr 474	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑦 ∈ ℂ)
8	4	negcld 10700	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 ∈ ℂ)
9	7	negcld 10700	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℂ)
10		1cnd 10351	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 1 ∈ ℂ)
11		simprr 791	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
12	11	difeq2d 3955	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) = (ℕ0 ∖ (bits‘𝑦)))
13		bitscmp 15533	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℤ → (ℕ0 ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
14	13	ad2antrr 719	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
15		bitscmp 15533	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℤ → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
16	15	ad2antlr 720	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
17	12, 14, 16	3eqtr3d 2869	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘(-𝑥 − 1)) = (bits‘(-𝑦 − 1)))
18		nnm1nn0 11661	. . . . . . . . . . 11 ⊢ (-𝑥 ∈ ℕ → (-𝑥 − 1) ∈ ℕ0)
19	18	ad2antrl 721	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) ∈ ℕ0)
20		fvres 6452	. . . . . . . . . 10 ⊢ ((-𝑥 − 1) ∈ ℕ0 → ((bits ↾ ℕ0)‘(-𝑥 − 1)) = (bits‘(-𝑥 − 1)))
21	19, 20	syl 17	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑥 − 1)) = (bits‘(-𝑥 − 1)))
22		ominf 8441	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ω ∈ Fin
23		nn0ennn 13073	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 ≈ ℕ
24		nnenom 13074	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ ≈ ω
25	23, 24	entr2i 8277	. . . . . . . . . . . . . . . . . 18 ⊢ ω ≈ ℕ0
26		enfii 8446	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℕ0 ∈ Fin ∧ ω ≈ ℕ0) → ω ∈ Fin)
27	25, 26	mpan2 684	. . . . . . . . . . . . . . . . 17 ⊢ (ℕ0 ∈ Fin → ω ∈ Fin)
28	22, 27	mto 189	. . . . . . . . . . . . . . . 16 ⊢ ¬ ℕ0 ∈ Fin
29		difinf 8499	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ℕ0 ∈ Fin ∧ (bits‘𝑥) ∈ Fin) → ¬ (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
30	28, 29	mpan 683	. . . . . . . . . . . . . . 15 ⊢ ((bits‘𝑥) ∈ Fin → ¬ (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
31		bitsfi 15532	. . . . . . . . . . . . . . . . 17 ⊢ ((-𝑥 − 1) ∈ ℕ0 → (bits‘(-𝑥 − 1)) ∈ Fin)
32	19, 31	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘(-𝑥 − 1)) ∈ Fin)
33	14, 32	eqeltrd 2906	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
34	30, 33	nsyl3 136	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑥) ∈ Fin)
35	11, 34	eqneltrrd 2926	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑦) ∈ Fin)
36		bitsfi 15532	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0 → (bits‘𝑦) ∈ Fin)
37	35, 36	nsyl 138	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ 𝑦 ∈ ℕ0)
38	5	znegcld 11812	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
39		elznn 11720	. . . . . . . . . . . . . . . . 17 ⊢ (-𝑦 ∈ ℤ ↔ (-𝑦 ∈ ℝ ∧ (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0)))
40	39	simprbi 492	. . . . . . . . . . . . . . . 16 ⊢ (-𝑦 ∈ ℤ → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0))
41	38, 40	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0))
42	6	negnegd 10704	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑦 = 𝑦)
43	42	eleq1d 2891	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑦 ∈ ℕ0 ↔ 𝑦 ∈ ℕ0))
44	43	orbi2d 946	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0) ↔ (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0)))
45	41, 44	mpbid 224	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
46	45	adantr 474	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
47	46	ord 897	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (¬ -𝑦 ∈ ℕ → 𝑦 ∈ ℕ0))
48	37, 47	mt3d 143	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℕ)
49		nnm1nn0 11661	. . . . . . . . . . 11 ⊢ (-𝑦 ∈ ℕ → (-𝑦 − 1) ∈ ℕ0)
50	48, 49	syl 17	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 − 1) ∈ ℕ0)
51		fvres 6452	. . . . . . . . . 10 ⊢ ((-𝑦 − 1) ∈ ℕ0 → ((bits ↾ ℕ0)‘(-𝑦 − 1)) = (bits‘(-𝑦 − 1)))
52	50, 51	syl 17	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑦 − 1)) = (bits‘(-𝑦 − 1)))
53	17, 21, 52	3eqtr4d 2871	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)))
54		bitsf1o 15540	. . . . . . . . . . 11 ⊢ (bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin)
55		f1of1 6377	. . . . . . . . . . 11 ⊢ ((bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin) → (bits ↾ ℕ0):ℕ0–1-1→(𝒫 ℕ0 ∩ Fin))
56	54, 55	ax-mp 5	. . . . . . . . . 10 ⊢ (bits ↾ ℕ0):ℕ0–1-1→(𝒫 ℕ0 ∩ Fin)
57		f1fveq 6774	. . . . . . . . . 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)))
58	56, 57	mpan 683	. . . . . . . . 9 ⊢ (((-𝑥 − 1) ∈ ℕ0 ∧ (-𝑦 − 1) ∈ ℕ0) → (((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
59	19, 50, 58	syl2anc 581	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
60	53, 59	mpbid 224	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) = (-𝑦 − 1))
61	8, 9, 10, 60	subcan2d 10755	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 = -𝑦)
62	4, 7, 61	neg11d 10725	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
63	62	expr 450	. . . 4 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ -𝑥 ∈ ℕ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
64	3	negnegd 10704	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑥 = 𝑥)
65	64	eleq1d 2891	. . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑥 ∈ ℕ0 ↔ 𝑥 ∈ ℕ0))
66	65	biimpa 470	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
67		simprr 791	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
68		fvres 6452	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → ((bits ↾ ℕ0)‘𝑥) = (bits‘𝑥))
69	68	ad2antrl 721	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑥) = (bits‘𝑥))
70	15	ad2antlr 720	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
71		bitsfi 15532	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → (bits‘𝑥) ∈ Fin)
72	71	ad2antrl 721	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) ∈ Fin)
73	67, 72	eqeltrrd 2907	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑦) ∈ Fin)
74		difinf 8499	. . . . . . . . . . . . . 14 ⊢ ((¬ ℕ0 ∈ Fin ∧ (bits‘𝑦) ∈ Fin) → ¬ (ℕ0 ∖ (bits‘𝑦)) ∈ Fin)
75	28, 73, 74	sylancr 583	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (ℕ0 ∖ (bits‘𝑦)) ∈ Fin)
76	70, 75	eqneltrrd 2926	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘(-𝑦 − 1)) ∈ Fin)
77		bitsfi 15532	. . . . . . . . . . . 12 ⊢ ((-𝑦 − 1) ∈ ℕ0 → (bits‘(-𝑦 − 1)) ∈ Fin)
78	76, 77	nsyl 138	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (-𝑦 − 1) ∈ ℕ0)
79	78, 49	nsyl 138	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ -𝑦 ∈ ℕ)
80	45	adantr 474	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
81	80	ord 897	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (¬ -𝑦 ∈ ℕ → 𝑦 ∈ ℕ0))
82	79, 81	mpd 15	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑦 ∈ ℕ0)
83		fvres 6452	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → ((bits ↾ ℕ0)‘𝑦) = (bits‘𝑦))
84	82, 83	syl 17	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑦) = (bits‘𝑦))
85	67, 69, 84	3eqtr4d 2871	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦))
86		simprl 789	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℕ0)
87		f1fveq 6774	. . . . . . . . 9 ⊢ (((bits ↾ ℕ0):ℕ0–1-1→(𝒫 ℕ0 ∩ Fin) ∧ (𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
88	56, 87	mpan 683	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
89	86, 82, 88	syl2anc 581	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
90	85, 89	mpbid 224	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
91	90	expr 450	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ 𝑥 ∈ ℕ0) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
92	66, 91	syldan 587	. . . 4 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ0) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
93	2	znegcld 11812	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑥 ∈ ℤ)
94		elznn 11720	. . . . . 6 ⊢ (-𝑥 ∈ ℤ ↔ (-𝑥 ∈ ℝ ∧ (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0)))
95	94	simprbi 492	. . . . 5 ⊢ (-𝑥 ∈ ℤ → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0))
96	93, 95	syl 17	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0))
97	63, 92, 96	mpjaodan 988	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
98	97	rgen2a 3186	. 2 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦)
99		dff13 6767	. 2 ⊢ (bits:ℤ–1-1→𝒫 ℕ0 ↔ (bits:ℤ⟶𝒫 ℕ0 ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦)))
100	1, 98, 99	mpbir2an 704	1 ⊢ bits:ℤ–1-1→𝒫 ℕ0

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 880   = wceq 1658   ∈ wcel 2166  ∀wral 3117   ∖ cdif 3795   ∩ cin 3797  𝒫 cpw 4378   class class class wbr 4873   ↾ cres 5344  ⟶wf 6119  –1-1→wf1 6120  –1-1-onto→wf1o 6122  ‘cfv 6123  (class class class)co 6905  ωcom 7326   ≈ cen 8219  Fincfn 8222  ℂcc 10250  ℝcr 10251  1c1 10253   − cmin 10585  -cneg 10586  ℕcn 11350  ℕ₀cn0 11618  ℤcz 11704  bitscbits 15514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-pre-sup 10330

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-disj 4842  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-sup 8617  df-inf 8618  df-oi 8684  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-nn 11351  df-2 11414  df-3 11415  df-n0 11619  df-xnn0 11691  df-z 11705  df-uz 11969  df-rp 12113  df-fz 12620  df-fzo 12761  df-fl 12888  df-mod 12964  df-seq 13096  df-exp 13155  df-hash 13411  df-cj 14216  df-re 14217  df-im 14218  df-sqrt 14352  df-abs 14353  df-clim 14596  df-sum 14794  df-dvds 15358  df-bits 15517

This theorem is referenced by:  bitsuz  15569  eulerpartlemmf  30982