Theorem wwlksnextinj 30320

Description: Lemma for wwlksnextbij 30323. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)

Hypotheses

Ref	Expression
wwlksnextbij0.v	⊢ 𝑉 = (Vtx‘𝐺)
wwlksnextbij0.e	⊢ 𝐸 = (Edg‘𝐺)
wwlksnextbij0.d	⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
wwlksnextbij0.r	⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij0.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))

Assertion

Ref	Expression
wwlksnextinj	⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷–1-1→𝑅)

Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑊   𝑡,𝐷   𝑛,𝐸   𝑤,𝐸   𝑡,𝑁,𝑤   𝑡,𝑅   𝑛,𝑉   𝑤,𝑉   𝑛,𝑊   𝑡,𝑛

Allowed substitution hints:   𝐷(𝑤, 𝑛)   𝑅(𝑤, 𝑛)   𝐸(𝑡)   𝐹(𝑤, 𝑡, 𝑛)   𝐺(𝑡, 𝑛)   𝑁(𝑛)   𝑉(𝑡)   𝑊(𝑡)

Proof of Theorem wwlksnextinj

Dummy variables 𝑑 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksnextbij0.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		wwlksnextbij0.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
3		wwlksnextbij0.d	. . 3 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
4		wwlksnextbij0.r	. . 3 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
5		wwlksnextbij0.f	. . 3 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))
6	1, 2, 3, 4, 5	wwlksnextfun 30319	. 2 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)
7		fveq2 6886	. . . . . . 7 ⊢ (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
8		fvex 6899	. . . . . . 7 ⊢ (lastS‘𝑑) ∈ V
9	7, 5, 8	fvmpt 6994	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 → (𝐹‘𝑑) = (lastS‘𝑑))
10		fveq2 6886	. . . . . . 7 ⊢ (𝑡 = 𝑥 → (lastS‘𝑡) = (lastS‘𝑥))
11		fvex 6899	. . . . . . 7 ⊢ (lastS‘𝑥) ∈ V
12	10, 5, 11	fvmpt 6994	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (lastS‘𝑥))
13	9, 12	eqeqan12d 2779	. . . . 5 ⊢ ((𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑑) = (𝐹‘𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
14	13	adantl 487	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((𝐹‘𝑑) = (𝐹‘𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
15		fveqeq2 6895	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑑) = (𝑁 + 2)))
16		oveq1 7427	. . . . . . . . 9 ⊢ (𝑤 = 𝑑 → (𝑤 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
17	16	eqeq1d 2767	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑑 prefix (𝑁 + 1)) = 𝑊))
18		fveq2 6886	. . . . . . . . . 10 ⊢ (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
19	18	preq2d 4708	. . . . . . . . 9 ⊢ (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
20	19	eleq1d 2850	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
21	15, 17, 20	3anbi123d 1464	. . . . . . 7 ⊢ (𝑤 = 𝑑 → (((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
22	21, 3	elrab2 3656	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 ↔ (𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
23		fveqeq2 6895	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑥) = (𝑁 + 2)))
24		oveq1 7427	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
25	24	eqeq1d 2767	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑥 prefix (𝑁 + 1)) = 𝑊))
26		fveq2 6886	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (lastS‘𝑤) = (lastS‘𝑥))
27	26	preq2d 4708	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑥)})
28	27	eleq1d 2850	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))
29	23, 25, 28	3anbi123d 1464	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)))
30	29, 3	elrab2 3656	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)))
31		eqtr3 2787	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (♯‘𝑥) = (𝑁 + 2)) → (♯‘𝑑) = (♯‘𝑥))
32	31	expcom 419	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑥) = (𝑁 + 2) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
33	32	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
34	33	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
35	34	com12 33	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑑) = (𝑁 + 2) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (♯‘𝑑) = (♯‘𝑥)))
36	35	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (♯‘𝑑) = (♯‘𝑥)))
37	36	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (♯‘𝑑) = (♯‘𝑥)))
38	37	imp 412	. . . . . . . . . 10 ⊢ (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (♯‘𝑑) = (♯‘𝑥))
39	38	adantr 486	. . . . . . . . 9 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → (♯‘𝑑) = (♯‘𝑥))
40	39	adantr 486	. . . . . . . 8 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (♯‘𝑑) = (♯‘𝑥))
41		simpr 490	. . . . . . . 8 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (lastS‘𝑑) = (lastS‘𝑥))
42		eqtr3 2787	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊) → (𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
43		1e2m1 12387	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 = (2 − 1)
44	43	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ0 → 1 = (2 − 1))
45	44	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) = (𝑁 + (2 − 1)))
46		nn0cn 12534	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
47		2cnd 12339	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
48		1cnd 11222	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
49	46, 47, 48	addsubassd 11609	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
50	45, 49	eqtr4d 2803	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) = ((𝑁 + 2) − 1))
51	50	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((𝑁 + 2) − 1))
52		oveq1 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑑) = (𝑁 + 2) → ((♯‘𝑑) − 1) = ((𝑁 + 2) − 1))
53	52	eqeq2d 2776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑑) = (𝑁 + 2) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
54	53	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
55	51, 54	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((♯‘𝑑) − 1))
56		oveq2 7428	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑑 prefix (𝑁 + 1)) = (𝑑 prefix ((♯‘𝑑) − 1)))
57		oveq2 7428	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))
58	56, 57	eqeq12d 2781	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) = ((♯‘𝑑) − 1) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ↔ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
59	55, 58	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ↔ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
60	59	biimpd 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
61	60	ex 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → ((♯‘𝑑) = (𝑁 + 2) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
62	61	com13 89	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
63	42, 62	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊) → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
64	63	ex 418	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 prefix (𝑁 + 1)) = 𝑊 → ((𝑥 prefix (𝑁 + 1)) = 𝑊 → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))))
65	64	com23 87	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 prefix (𝑁 + 1)) = 𝑊 → ((♯‘𝑑) = (𝑁 + 2) → ((𝑥 prefix (𝑁 + 1)) = 𝑊 → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))))
66	65	impcom 413	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → ((𝑥 prefix (𝑁 + 1)) = 𝑊 → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
67	66	com12 33	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 prefix (𝑁 + 1)) = 𝑊 → (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
68	67	3ad2ant2 1152	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
69	68	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
70	69	com12 33	. . . . . . . . . . . 12 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
71	70	3adant3 1150	. . . . . . . . . . 11 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
72	71	adantl 487	. . . . . . . . . 10 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
73	72	imp31 423	. . . . . . . . 9 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))
74	73	adantr 486	. . . . . . . 8 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))
75		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → 𝑑 ∈ Word 𝑉)
76		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → 𝑥 ∈ Word 𝑉)
77	75, 76	anim12i 625	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉))
78	77	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → (𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉))
79		nn0re 12533	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
80		2re 12335	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℝ
81	80	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
82		nn0ge0 12549	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
83		2pos 12365	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 < 2
84	83	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 0 < 2)
85	79, 81, 82, 84	addgegt0d 11807	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
86	85	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
87		breq2 5115	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑑) = (𝑁 + 2) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
88	87	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
89	86, 88	mpbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (♯‘𝑑))
90		hashgt0n0 14424	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 0 < (♯‘𝑑)) → 𝑑 ≠ ∅)
91	89, 90	sylan2 605	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑑 ≠ ∅)
92	91	exp32 426	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ Word 𝑉 → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → 𝑑 ≠ ∅)))
93	92	com12 33	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑑) = (𝑁 + 2) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → 𝑑 ≠ ∅)))
94	93	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → 𝑑 ≠ ∅)))
95	94	impcom 413	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → 𝑑 ≠ ∅))
96	95	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ0 → 𝑑 ≠ ∅))
97	96	imp 412	. . . . . . . . . . 11 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → 𝑑 ≠ ∅)
98	85	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
99		breq2 5115	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = (𝑁 + 2) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
100	99	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
101	98, 100	mpbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (♯‘𝑥))
102		hashgt0n0 14424	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ Word 𝑉 ∧ 0 < (♯‘𝑥)) → 𝑥 ≠ ∅)
103	101, 102	sylan2 605	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑥 ≠ ∅)
104	103	exp32 426	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word 𝑉 → ((♯‘𝑥) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → 𝑥 ≠ ∅)))
105	104	com12 33	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = (𝑁 + 2) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → 𝑥 ≠ ∅)))
106	105	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → 𝑥 ≠ ∅)))
107	106	impcom 413	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → 𝑥 ≠ ∅))
108	107	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ0 → 𝑥 ≠ ∅))
109	108	imp 412	. . . . . . . . . . 11 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → 𝑥 ≠ ∅)
110	78, 97, 109	jca32 525	. . . . . . . . . 10 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
111	110	adantr 486	. . . . . . . . 9 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
112		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 𝑑 ∈ Word 𝑉)
113	112	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑑 ∈ Word 𝑉)
114		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 𝑥 ∈ Word 𝑉)
115	114	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ Word 𝑉)
116		hashneq0 14423	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ Word 𝑉 → (0 < (♯‘𝑑) ↔ 𝑑 ≠ ∅))
117	116	biimprd 251	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ Word 𝑉 → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
118	117	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
119	118	com12 33	. . . . . . . . . . . . 13 ⊢ (𝑑 ≠ ∅ → ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
120	119	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅) → ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
121	120	impcom 413	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 0 < (♯‘𝑑))
122		pfxsuff1eqwrdeq 14763	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉 ∧ 0 < (♯‘𝑑)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥)))))
123	113, 115, 121, 122	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥)))))
124		ancom 466	. . . . . . . . . . . 12 ⊢ (((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥)) ↔ ((lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
125	124	anbi2i 635	. . . . . . . . . . 11 ⊢ (((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥))) ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
126		3anass 1111	. . . . . . . . . . 11 ⊢ (((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))) ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
127	125, 126	bitr4i 281	. . . . . . . . . 10 ⊢ (((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥))) ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
128	123, 127	bitrdi 290	. . . . . . . . 9 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
129	111, 128	syl 18	. . . . . . . 8 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
130	40, 41, 74, 129	mpbir3and 1361	. . . . . . 7 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → 𝑑 = 𝑥)
131	130	exp31 425	. . . . . 6 ⊢ (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ0 → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥)))
132	22, 30, 131	syl2anb 610	. . . . 5 ⊢ ((𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷) → (𝑁 ∈ ℕ0 → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥)))
133	132	impcom 413	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥))
134	14, 133	sylbid 243	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥))
135	134	ralrimivva 3210	. 2 ⊢ (𝑁 ∈ ℕ0 → ∀𝑑 ∈ 𝐷 ∀𝑥 ∈ 𝐷 ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥))
136		dff13 7258	. 2 ⊢ (𝐹:𝐷–1-1→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑑 ∈ 𝐷 ∀𝑥 ∈ 𝐷 ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥)))
137	6, 135, 136	sylanbrc 595	1 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷–1-1→𝑅)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  {crab 3418  ∅c0 4286  {cpr 4593   class class class wbr 5111   ↦ cmpt 5194  ⟶wf 6537  –1-1→wf1 6538  ‘cfv 6541  (class class class)co 7420  ℝcr 11119  0cc0 11120  1c1 11121   + caddc 11123   < clt 11263   − cmin 11461  2c2 12315  ℕ₀cn0 12524  ♯chash 14389  Word cword 14573  lastSclsw 14622   prefix cpfx 14735  Vtxcvtx 29406  Edgcedg 29457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7743  ax-cnex 11176  ax-resscn 11177  ax-1cn 11178  ax-icn 11179  ax-addcl 11180  ax-addrcl 11181  ax-mulcl 11182  ax-mulrcl 11183  ax-mulcom 11184  ax-addass 11185  ax-mulass 11186  ax-distr 11187  ax-i2m1 11188  ax-1ne0 11189  ax-1rid 11190  ax-rnegex 11191  ax-rrecex 11192  ax-cnre 11193  ax-pre-lttri 11194  ax-pre-lttrn 11195  ax-pre-ltadd 11196  ax-pre-mulgt0 11197

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7870  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8460  df-er 8701  df-en 8951  df-dom 8952  df-sdom 8953  df-fin 8954  df-card 9942  df-pnf 11265  df-mnf 11266  df-xr 11267  df-ltxr 11268  df-le 11269  df-sub 11463  df-neg 11464  df-nn 12254  df-2 12323  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12884  df-fz 13557  df-fzo 13705  df-hash 14390  df-word 14574  df-lsw 14623  df-s1 14658  df-substr 14704  df-pfx 14736

This theorem is used by:  wwlksnextbij0  30322