Theorem wwlktovf1 14672

Description: Lemma 2 for wrd2f1tovbij 14675. (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 14671	. 2 ⊢ 𝐹:𝐷⟶𝑅
5		fveq1 6773	. . . . . 6 ⊢ (𝑡 = 𝑥 → (𝑡‘1) = (𝑥‘1))
6		fvex 6787	. . . . . 6 ⊢ (𝑥‘1) ∈ V
7	5, 3, 6	fvmpt 6875	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (𝑥‘1))
8		fveq1 6773	. . . . . 6 ⊢ (𝑡 = 𝑦 → (𝑡‘1) = (𝑦‘1))
9		fvex 6787	. . . . . 6 ⊢ (𝑦‘1) ∈ V
10	8, 3, 9	fvmpt 6875	. . . . 5 ⊢ (𝑦 ∈ 𝐷 → (𝐹‘𝑦) = (𝑦‘1))
11	7, 10	eqeqan12d 2752	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥‘1) = (𝑦‘1)))
12		fveqeq2 6783	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((♯‘𝑤) = 2 ↔ (♯‘𝑥) = 2))
13		fveq1 6773	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
14	13	eqeq1d 2740	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((𝑤‘0) = 𝑃 ↔ (𝑥‘0) = 𝑃))
15		fveq1 6773	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤‘1) = (𝑥‘1))
16	13, 15	preq12d 4677	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → {(𝑤‘0), (𝑤‘1)} = {(𝑥‘0), (𝑥‘1)})
17	16	eleq1d 2823	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋))
18	12, 14, 17	3anbi123d 1435	. . . . . 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 3627	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
20		fveqeq2 6783	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((♯‘𝑤) = 2 ↔ (♯‘𝑦) = 2))
21		fveq1 6773	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
22	21	eqeq1d 2740	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
23		fveq1 6773	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤‘1) = (𝑦‘1))
24	21, 23	preq12d 4677	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → {(𝑤‘0), (𝑤‘1)} = {(𝑦‘0), (𝑦‘1)})
25	24	eleq1d 2823	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))
26	20, 22, 25	3anbi123d 1435	. . . . . 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 3627	. . . . 5 ⊢ (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
28		simpr1 1193	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (♯‘𝑥) = 2)
29		simpr1 1193	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → (♯‘𝑦) = 2)
30	29	eqcomd 2744	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 2 = (♯‘𝑦))
31	28, 30	sylan9eq 2798	. . . . . . . 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 481	. . . . . . 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 1194	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (𝑥‘0) = 𝑃)
34		simpr2 1194	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → (𝑦‘0) = 𝑃)
35	34	eqcomd 2744	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 𝑃 = (𝑦‘0))
36	33, 35	sylan9eq 2798	. . . . . . . . 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 481	. . . . . . . 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 485	. . . . . . . 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 7283	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = (0..^2))
40		fzo0to2pr 13472	. . . . . . . . . . . . 13 ⊢ (0..^2) = {0, 1}
41	39, 40	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = {0, 1})
42	41	raleqdv 3348	. . . . . . . . . . 11 ⊢ ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ∀𝑖 ∈ {0, 1} (𝑥‘𝑖) = (𝑦‘𝑖)))
43		c0ex 10969	. . . . . . . . . . . 12 ⊢ 0 ∈ V
44		1ex 10971	. . . . . . . . . . . 12 ⊢ 1 ∈ V
45		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑥‘𝑖) = (𝑥‘0))
46		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑦‘𝑖) = (𝑦‘0))
47	45, 46	eqeq12d 2754	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
48		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑥‘𝑖) = (𝑥‘1))
49		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑦‘𝑖) = (𝑦‘1))
50	48, 49	eqeq12d 2754	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘1) = (𝑦‘1)))
51	43, 44, 47, 50	ralpr 4636	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ {0, 1} (𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1)))
52	42, 51	bitrdi 287	. . . . . . . . . 10 ⊢ ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
53	52	3ad2ant1 1132	. . . . . . . . 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 728	. . . . . . . 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 710	. . . . . . 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 14260	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖))))
57	56	ad2ant2r 744	. . . . . . . 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 481	. . . . . . 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 710	. . . . . 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 413	. . . . 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 598	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
62	11, 61	sylbid 239	. . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
63	62	rgen2 3120	. 2 ⊢ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
64		dff13 7128	. 2 ⊢ (𝐹:𝐷–1-1→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
65	4, 63, 64	mpbir2an 708	1 ⊢ 𝐹:𝐷–1-1→𝑅

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  {cpr 4563   ↦ cmpt 5157  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433  (class class class)co 7275  0cc0 10871  1c1 10872  2c2 12028  ..^cfzo 13382  ♯chash 14044  Word cword 14217

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-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

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-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-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-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-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218

This theorem is referenced by:  wwlktovf1o  14674