Theorem wwlktovf1 14368

Description: Lemma 2 for wrd2f1tovbij 14371. (Contributed by Alexander van der Vekens, 27-Jul-2018.)

Hypotheses

Ref	Expression
wwlktovf1o.d	⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
wwlktovf1o.r	⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
wwlktovf1o.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1))

Assertion

Ref	Expression
wwlktovf1	⊢ 𝐹:𝐷–1-1→𝑅

Distinct variable groups:   𝑡,𝐷   𝑃,𝑛,𝑡,𝑤   𝑡,𝑅   𝑛,𝑉,𝑡,𝑤   𝑛,𝑋,𝑤

Allowed substitution hints:   𝐷(𝑤,𝑛)   𝑅(𝑤,𝑛)   𝐹(𝑤,𝑡,𝑛)   𝑋(𝑡)

Proof of Theorem wwlktovf1

Dummy variables 𝑥 𝑦 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlktovf1o.d	. . 3 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
2		wwlktovf1o.r	. . 3 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}
3		wwlktovf1o.f	. . 3 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1))
4	1, 2, 3	wwlktovf 14367	. 2 ⊢ 𝐹:𝐷⟶𝑅
5		fveq1 6657	. . . . . 6 ⊢ (𝑡 = 𝑥 → (𝑡‘1) = (𝑥‘1))
6		fvex 6671	. . . . . 6 ⊢ (𝑥‘1) ∈ V
7	5, 3, 6	fvmpt 6759	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (𝑥‘1))
8		fveq1 6657	. . . . . 6 ⊢ (𝑡 = 𝑦 → (𝑡‘1) = (𝑦‘1))
9		fvex 6671	. . . . . 6 ⊢ (𝑦‘1) ∈ V
10	8, 3, 9	fvmpt 6759	. . . . 5 ⊢ (𝑦 ∈ 𝐷 → (𝐹‘𝑦) = (𝑦‘1))
11	7, 10	eqeqan12d 2775	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥‘1) = (𝑦‘1)))
12		fveqeq2 6667	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((♯‘𝑤) = 2 ↔ (♯‘𝑥) = 2))
13		fveq1 6657	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
14	13	eqeq1d 2760	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((𝑤‘0) = 𝑃 ↔ (𝑥‘0) = 𝑃))
15		fveq1 6657	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤‘1) = (𝑥‘1))
16	13, 15	preq12d 4634	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → {(𝑤‘0), (𝑤‘1)} = {(𝑥‘0), (𝑥‘1)})
17	16	eleq1d 2836	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋))
18	12, 14, 17	3anbi123d 1433	. . . . . 6 ⊢ (𝑤 = 𝑥 → (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
19	18, 1	elrab2 3605	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
20		fveqeq2 6667	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((♯‘𝑤) = 2 ↔ (♯‘𝑦) = 2))
21		fveq1 6657	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
22	21	eqeq1d 2760	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
23		fveq1 6657	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤‘1) = (𝑦‘1))
24	21, 23	preq12d 4634	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → {(𝑤‘0), (𝑤‘1)} = {(𝑦‘0), (𝑦‘1)})
25	24	eleq1d 2836	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))
26	20, 22, 25	3anbi123d 1433	. . . . . 6 ⊢ (𝑤 = 𝑦 → (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
27	26, 1	elrab2 3605	. . . . 5 ⊢ (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
28		simpr1 1191	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (♯‘𝑥) = 2)
29		simpr1 1191	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → (♯‘𝑦) = 2)
30	29	eqcomd 2764	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 2 = (♯‘𝑦))
31	28, 30	sylan9eq 2813	. . . . . . . 8 ⊢ (((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (♯‘𝑥) = (♯‘𝑦))
32	31	adantr 484	. . . . . . 7 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (♯‘𝑥) = (♯‘𝑦))
33		simpr2 1192	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (𝑥‘0) = 𝑃)
34		simpr2 1192	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → (𝑦‘0) = 𝑃)
35	34	eqcomd 2764	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 𝑃 = (𝑦‘0))
36	33, 35	sylan9eq 2813	. . . . . . . . 9 ⊢ (((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (𝑥‘0) = (𝑦‘0))
37	36	adantr 484	. . . . . . . 8 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥‘0) = (𝑦‘0))
38		simpr 488	. . . . . . . 8 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥‘1) = (𝑦‘1))
39		oveq2 7158	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = (0..^2))
40		fzo0to2pr 13171	. . . . . . . . . . . . 13 ⊢ (0..^2) = {0, 1}
41	39, 40	eqtrdi 2809	. . . . . . . . . . . 12 ⊢ ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = {0, 1})
42	41	raleqdv 3329	. . . . . . . . . . 11 ⊢ ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ∀𝑖 ∈ {0, 1} (𝑥‘𝑖) = (𝑦‘𝑖)))
43		c0ex 10673	. . . . . . . . . . . 12 ⊢ 0 ∈ V
44		1ex 10675	. . . . . . . . . . . 12 ⊢ 1 ∈ V
45		fveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑥‘𝑖) = (𝑥‘0))
46		fveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑦‘𝑖) = (𝑦‘0))
47	45, 46	eqeq12d 2774	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
48		fveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑥‘𝑖) = (𝑥‘1))
49		fveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑦‘𝑖) = (𝑦‘1))
50	48, 49	eqeq12d 2774	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘1) = (𝑦‘1)))
51	43, 44, 47, 50	ralpr 4593	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ {0, 1} (𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1)))
52	42, 51	bitrdi 290	. . . . . . . . . 10 ⊢ ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
53	52	3ad2ant1 1130	. . . . . . . . 9 ⊢ (((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋) → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
54	53	ad3antlr 730	. . . . . . . 8 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
55	37, 38, 54	mpbir2and 712	. . . . . . 7 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖))
56		eqwrd 13956	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖))))
57	56	ad2ant2r 746	. . . . . . . 8 ⊢ (((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖))))
58	57	adantr 484	. . . . . . 7 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖))))
59	32, 55, 58	mpbir2and 712	. . . . . 6 ⊢ ((((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) ∧ (𝑥‘1) = (𝑦‘1)) → 𝑥 = 𝑦)
60	59	ex 416	. . . . 5 ⊢ (((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) ∧ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
61	19, 27, 60	syl2anb 600	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
62	11, 61	sylbid 243	. . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
63	62	rgen2 3132	. 2 ⊢ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
64		dff13 7005	. 2 ⊢ (𝐹:𝐷–1-1→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
65	4, 63, 64	mpbir2an 710	1 ⊢ 𝐹:𝐷–1-1→𝑅

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  {cpr 4524   ↦ cmpt 5112  ⟶wf 6331  –1-1→wf1 6332  ‘cfv 6335  (class class class)co 7150  0cc0 10575  1c1 10576  2c2 11729  ..^cfzo 13082  ♯chash 13740  Word cword 13913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-n0 11935  df-z 12021  df-uz 12283  df-fz 12940  df-fzo 13083  df-hash 13741  df-word 13914

This theorem is referenced by:  wwlktovf1o  14370