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Theorem wwlksnextinj 27685
 Description: Lemma for wwlksnextbij 27688. (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 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij0.f 𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))
Assertion
Ref Expression
wwlksnextinj (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)
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.
StepHypRef 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 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
5 wwlksnextbij0.f . . 3 𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))
61, 2, 3, 4, 5wwlksnextfun 27684 . 2 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
7 fveq2 6645 . . . . . . 7 (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
8 fvex 6658 . . . . . . 7 (lastS‘𝑑) ∈ V
97, 5, 8fvmpt 6745 . . . . . 6 (𝑑𝐷 → (𝐹𝑑) = (lastS‘𝑑))
10 fveq2 6645 . . . . . . 7 (𝑡 = 𝑥 → (lastS‘𝑡) = (lastS‘𝑥))
11 fvex 6658 . . . . . . 7 (lastS‘𝑥) ∈ V
1210, 5, 11fvmpt 6745 . . . . . 6 (𝑥𝐷 → (𝐹𝑥) = (lastS‘𝑥))
139, 12eqeqan12d 2815 . . . . 5 ((𝑑𝐷𝑥𝐷) → ((𝐹𝑑) = (𝐹𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
1413adantl 485 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((𝐹𝑑) = (𝐹𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
15 fveqeq2 6654 . . . . . . . 8 (𝑤 = 𝑑 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑑) = (𝑁 + 2)))
16 oveq1 7142 . . . . . . . . 9 (𝑤 = 𝑑 → (𝑤 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
1716eqeq1d 2800 . . . . . . . 8 (𝑤 = 𝑑 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑑 prefix (𝑁 + 1)) = 𝑊))
18 fveq2 6645 . . . . . . . . . 10 (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
1918preq2d 4636 . . . . . . . . 9 (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
2019eleq1d 2874 . . . . . . . 8 (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
2115, 17, 203anbi123d 1433 . . . . . . 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)))
2221, 3elrab2 3631 . . . . . 6 (𝑑𝐷 ↔ (𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
23 fveqeq2 6654 . . . . . . . 8 (𝑤 = 𝑥 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑥) = (𝑁 + 2)))
24 oveq1 7142 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
2524eqeq1d 2800 . . . . . . . 8 (𝑤 = 𝑥 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑥 prefix (𝑁 + 1)) = 𝑊))
26 fveq2 6645 . . . . . . . . . 10 (𝑤 = 𝑥 → (lastS‘𝑤) = (lastS‘𝑥))
2726preq2d 4636 . . . . . . . . 9 (𝑤 = 𝑥 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑥)})
2827eleq1d 2874 . . . . . . . 8 (𝑤 = 𝑥 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))
2923, 25, 283anbi123d 1433 . . . . . . 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)))
3029, 3elrab2 3631 . . . . . 6 (𝑥𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)))
31 eqtr3 2820 . . . . . . . . . . . . . . . . 17 (((♯‘𝑑) = (𝑁 + 2) ∧ (♯‘𝑥) = (𝑁 + 2)) → (♯‘𝑑) = (♯‘𝑥))
3231expcom 417 . . . . . . . . . . . . . . . 16 ((♯‘𝑥) = (𝑁 + 2) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
33323ad2ant1 1130 . . . . . . . . . . . . . . 15 (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
3433adantl 485 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
3534com12 32 . . . . . . . . . . . . 13 ((♯‘𝑑) = (𝑁 + 2) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (♯‘𝑑) = (♯‘𝑥)))
36353ad2ant1 1130 . . . . . . . . . . . 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)))
3736adantl 485 . . . . . . . . . . 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)))
3837imp 410 . . . . . . . . . 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))
3938adantr 484 . . . . . . . . 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))
4039adantr 484 . . . . . . . 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 488 . . . . . . . 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 2820 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊) → (𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
43 1e2m1 11752 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 = (2 − 1)
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 1 = (2 − 1))
4544oveq2d 7151 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0 → (𝑁 + 1) = (𝑁 + (2 − 1)))
46 nn0cn 11895 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
47 2cnd 11703 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
48 1cnd 10625 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
4946, 47, 48addsubassd 11006 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
5045, 49eqtr4d 2836 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0 → (𝑁 + 1) = ((𝑁 + 2) − 1))
5150adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((𝑁 + 2) − 1))
52 oveq1 7142 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝑑) = (𝑁 + 2) → ((♯‘𝑑) − 1) = ((𝑁 + 2) − 1))
5352eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝑑) = (𝑁 + 2) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
5453adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
5551, 54mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((♯‘𝑑) − 1))
56 oveq2 7143 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑑 prefix (𝑁 + 1)) = (𝑑 prefix ((♯‘𝑑) − 1)))
57 oveq2 7143 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))
5856, 57eqeq12d 2814 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) = ((♯‘𝑑) − 1) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ↔ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
5955, 58syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ↔ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
6059biimpd 232 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
6160ex 416 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → ((♯‘𝑑) = (𝑁 + 2) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
6261com13 88 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
6342, 62syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊) → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
6463ex 416 . . . . . . . . . . . . . . . . . 18 ((𝑑 prefix (𝑁 + 1)) = 𝑊 → ((𝑥 prefix (𝑁 + 1)) = 𝑊 → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))))
6564com23 86 . . . . . . . . . . . . . . . . 17 ((𝑑 prefix (𝑁 + 1)) = 𝑊 → ((♯‘𝑑) = (𝑁 + 2) → ((𝑥 prefix (𝑁 + 1)) = 𝑊 → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))))
6665impcom 411 . . . . . . . . . . . . . . . 16 (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → ((𝑥 prefix (𝑁 + 1)) = 𝑊 → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
6766com12 32 . . . . . . . . . . . . . . 15 ((𝑥 prefix (𝑁 + 1)) = 𝑊 → (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
68673ad2ant2 1131 . . . . . . . . . . . . . 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)))))
6968adantl 485 . . . . . . . . . . . . 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)))))
7069com12 32 . . . . . . . . . . . 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)))))
71703adant3 1129 . . . . . . . . . . 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)))))
7271adantl 485 . . . . . . . . . 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)))))
7372imp31 421 . . . . . . . . 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)))
7473adantr 484 . . . . . . . 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 486 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → 𝑑 ∈ Word 𝑉)
76 simpl 486 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → 𝑥 ∈ Word 𝑉)
7775, 76anim12i 615 . . . . . . . . . . . 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 𝑉𝑥 ∈ Word 𝑉))
7877adantr 484 . . . . . . . . . . 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 𝑉𝑥 ∈ Word 𝑉))
79 nn0re 11894 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
80 2re 11699 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℝ
8180a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
82 nn0ge0 11910 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
83 2pos 11728 . . . . . . . . . . . . . . . . . . . . . 22 0 < 2
8483a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 < 2)
8579, 81, 82, 84addgegt0d 11202 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
8685adantl 485 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
87 breq2 5034 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑑) = (𝑁 + 2) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
8887adantr 484 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
8986, 88mpbird 260 . . . . . . . . . . . . . . . . . 18 (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (♯‘𝑑))
90 hashgt0n0 13722 . . . . . . . . . . . . . . . . . 18 ((𝑑 ∈ Word 𝑉 ∧ 0 < (♯‘𝑑)) → 𝑑 ≠ ∅)
9189, 90sylan2 595 . . . . . . . . . . . . . . . . 17 ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑑 ≠ ∅)
9291exp32 424 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
9392com12 32 . . . . . . . . . . . . . . 15 ((♯‘𝑑) = (𝑁 + 2) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
94933ad2ant1 1130 . . . . . . . . . . . . . 14 (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
9594impcom 411 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → (𝑁 ∈ ℕ0𝑑 ≠ ∅))
9695adantr 484 . . . . . . . . . . . 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 ∈ ℕ0𝑑 ≠ ∅))
9796imp 410 . . . . . . . . . . 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 ≠ ∅)
9885adantl 485 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
99 breq2 5034 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑥) = (𝑁 + 2) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
10099adantr 484 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
10198, 100mpbird 260 . . . . . . . . . . . . . . . . . 18 (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (♯‘𝑥))
102 hashgt0n0 13722 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ Word 𝑉 ∧ 0 < (♯‘𝑥)) → 𝑥 ≠ ∅)
103101, 102sylan2 595 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑥 ≠ ∅)
104103exp32 424 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Word 𝑉 → ((♯‘𝑥) = (𝑁 + 2) → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
105104com12 32 . . . . . . . . . . . . . . 15 ((♯‘𝑥) = (𝑁 + 2) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
1061053ad2ant1 1130 . . . . . . . . . . . . . 14 (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
107106impcom 411 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0𝑥 ≠ ∅))
108107adantl 485 . . . . . . . . . . . 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 ∈ ℕ0𝑥 ≠ ∅))
109108imp 410 . . . . . . . . . . 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 ≠ ∅)
11078, 97, 109jca32 519 . . . . . . . . . 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 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
111110adantr 484 . . . . . . . . 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 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
112 simpl 486 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 𝑑 ∈ Word 𝑉)
113112adantr 484 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑑 ∈ Word 𝑉)
114 simpr 488 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 𝑥 ∈ Word 𝑉)
115114adantr 484 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ Word 𝑉)
116 hashneq0 13721 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 → (0 < (♯‘𝑑) ↔ 𝑑 ≠ ∅))
117116biimprd 251 . . . . . . . . . . . . . . 15 (𝑑 ∈ Word 𝑉 → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
118117adantr 484 . . . . . . . . . . . . . 14 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
119118com12 32 . . . . . . . . . . . . 13 (𝑑 ≠ ∅ → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
120119adantr 484 . . . . . . . . . . . 12 ((𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅) → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
121120impcom 411 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 0 < (♯‘𝑑))
122 pfxsuff1eqwrdeq 14052 . . . . . . . . . . 11 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉 ∧ 0 < (♯‘𝑑)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥)))))
123113, 115, 121, 122syl3anc 1368 . . . . . . . . . 10 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥)))))
124 ancom 464 . . . . . . . . . . . 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))))
125124anbi2i 625 . . . . . . . . . . 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 1092 . . . . . . . . . . 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)))))
127125, 126bitr4i 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))))
128123, 127syl6bb 290 . . . . . . . . 9 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
129111, 128syl 17 . . . . . . . 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)))))
13040, 41, 74, 129mpbir3and 1339 . . . . . . 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)
131130exp31 423 . . . . . 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)))
13222, 30, 131syl2anb 600 . . . . 5 ((𝑑𝐷𝑥𝐷) → (𝑁 ∈ ℕ0 → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥)))
133132impcom 411 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥))
13414, 133sylbid 243 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥))
135134ralrimivva 3156 . 2 (𝑁 ∈ ℕ0 → ∀𝑑𝐷𝑥𝐷 ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥))
136 dff13 6991 . 2 (𝐹:𝐷1-1𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑑𝐷𝑥𝐷 ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥)))
1376, 135, 136sylanbrc 586 1 (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  {crab 3110  ∅c0 4243  {cpr 4527   class class class wbr 5030   ↦ cmpt 5110  ⟶wf 6320  –1-1→wf1 6321  ‘cfv 6324  (class class class)co 7135  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529   < clt 10664   − cmin 10859  2c2 11680  ℕ0cn0 11885  ♯chash 13686  Word cword 13857  lastSclsw 13905   prefix cpfx 14023  Vtxcvtx 26789  Edgcedg 26840 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 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-lsw 13906  df-s1 13941  df-substr 13994  df-pfx 14024 This theorem is referenced by:  wwlksnextbij0  27687
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