Theorem wwlktovf1 14914

Description: Lemma 2 for wrd2f1tovbij 14917. (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 14913	. 2 ⊢ 𝐹:𝐷⟶𝑅
5		fveq1 6830	. . . . . 6 ⊢ (𝑡 = 𝑥 → (𝑡‘1) = (𝑥‘1))
6		fvex 6844	. . . . . 6 ⊢ (𝑥‘1) ∈ V
7	5, 3, 6	fvmpt 6939	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (𝑥‘1))
8		fveq1 6830	. . . . . 6 ⊢ (𝑡 = 𝑦 → (𝑡‘1) = (𝑦‘1))
9		fvex 6844	. . . . . 6 ⊢ (𝑦‘1) ∈ V
10	8, 3, 9	fvmpt 6939	. . . . 5 ⊢ (𝑦 ∈ 𝐷 → (𝐹‘𝑦) = (𝑦‘1))
11	7, 10	eqeqan12d 2755	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥‘1) = (𝑦‘1)))
12		fveqeq2 6840	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((♯‘𝑤) = 2 ↔ (♯‘𝑥) = 2))
13		fveq1 6830	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
14	13	eqeq1d 2743	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((𝑤‘0) = 𝑃 ↔ (𝑥‘0) = 𝑃))
15		fveq1 6830	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤‘1) = (𝑥‘1))
16	13, 15	preq12d 4676	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → {(𝑤‘0), (𝑤‘1)} = {(𝑥‘0), (𝑥‘1)})
17	16	eleq1d 2826	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋))
18	12, 14, 17	3anbi123d 1445	. . . . . 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 3634	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)))
20		fveqeq2 6840	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((♯‘𝑤) = 2 ↔ (♯‘𝑦) = 2))
21		fveq1 6830	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
22	21	eqeq1d 2743	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
23		fveq1 6830	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤‘1) = (𝑦‘1))
24	21, 23	preq12d 4676	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → {(𝑤‘0), (𝑤‘1)} = {(𝑦‘0), (𝑦‘1)})
25	24	eleq1d 2826	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋))
26	20, 22, 25	3anbi123d 1445	. . . . . 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 3634	. . . . 5 ⊢ (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)))
28		simpr1 1202	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (♯‘𝑥) = 2)
29		simpr1 1202	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → (♯‘𝑦) = 2)
30	29	eqcomd 2747	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 2 = (♯‘𝑦))
31	28, 30	sylan9eq 2796	. . . . . . . 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 482	. . . . . . 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 1203	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = 2 ∧ (𝑥‘0) = 𝑃 ∧ {(𝑥‘0), (𝑥‘1)} ∈ 𝑋)) → (𝑥‘0) = 𝑃)
34		simpr2 1203	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → (𝑦‘0) = 𝑃)
35	34	eqcomd 2747	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Word 𝑉 ∧ ((♯‘𝑦) = 2 ∧ (𝑦‘0) = 𝑃 ∧ {(𝑦‘0), (𝑦‘1)} ∈ 𝑋)) → 𝑃 = (𝑦‘0))
36	33, 35	sylan9eq 2796	. . . . . . . . 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 482	. . . . . . . 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 486	. . . . . . . 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 7368	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = (0..^2))
40		fzo0to2pr 13700	. . . . . . . . . . . . 13 ⊢ (0..^2) = {0, 1}
41	39, 40	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ ((♯‘𝑥) = 2 → (0..^(♯‘𝑥)) = {0, 1})
42	41	raleqdv 3299	. . . . . . . . . . 11 ⊢ ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ∀𝑖 ∈ {0, 1} (𝑥‘𝑖) = (𝑦‘𝑖)))
43		c0ex 11133	. . . . . . . . . . . 12 ⊢ 0 ∈ V
44		1ex 11135	. . . . . . . . . . . 12 ⊢ 1 ∈ V
45		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑥‘𝑖) = (𝑥‘0))
46		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑦‘𝑖) = (𝑦‘0))
47	45, 46	eqeq12d 2757	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
48		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑥‘𝑖) = (𝑥‘1))
49		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑦‘𝑖) = (𝑦‘1))
50	48, 49	eqeq12d 2757	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘1) = (𝑦‘1)))
51	43, 44, 47, 50	ralpr 4635	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ {0, 1} (𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1)))
52	42, 51	bitrdi 289	. . . . . . . . . 10 ⊢ ((♯‘𝑥) = 2 → (∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ((𝑥‘0) = (𝑦‘0) ∧ (𝑥‘1) = (𝑦‘1))))
53	52	3ad2ant1 1140	. . . . . . . . 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 738	. . . . . . . 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 720	. . . . . . 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 14514	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ∀𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) = (𝑦‘𝑖))))
57	56	ad2ant2r 754	. . . . . . . 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 482	. . . . . . 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 720	. . . . . 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 414	. . . . 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 605	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝑥‘1) = (𝑦‘1) → 𝑥 = 𝑦))
62	11, 61	sylbid 242	. . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
63	62	rgen2 3181	. 2 ⊢ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
64		dff13 7202	. 2 ⊢ (𝐹:𝐷–1-1→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
65	4, 63, 64	mpbir2an 718	1 ⊢ 𝐹:𝐷–1-1→𝑅

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  {crab 3393  {cpr 4560   ↦ cmpt 5156  ⟶wf 6485  –1-1→wf1 6486  ‘cfv 6489  (class class class)co 7360  0cc0 11033  1c1 11034  2c2 12231  ..^cfzo 13603  ♯chash 14287  Word cword 14470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-n0 12433  df-z 12520  df-uz 12784  df-fz 13457  df-fzo 13604  df-hash 14288  df-word 14471

This theorem is referenced by:  wwlktovf1o  14916