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Theorem wwlksnextinj 26862
 Description: Lemma for wwlksnextbij 26865. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.)
Hypotheses
Ref Expression
wwlksnextbij0.v 𝑉 = (Vtx‘𝐺)
wwlksnextbij0.e 𝐸 = (Edg‘𝐺)
wwlksnextbij0.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
wwlksnextbij.r 𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij.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 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
4 wwlksnextbij.r . . 3 𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}
5 wwlksnextbij.f . . 3 𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))
61, 2, 3, 4, 5wwlksnextfun 26861 . 2 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
7 fveq2 6229 . . . . . . 7 (𝑡 = 𝑑 → ( lastS ‘𝑡) = ( lastS ‘𝑑))
8 fvex 6239 . . . . . . 7 ( lastS ‘𝑑) ∈ V
97, 5, 8fvmpt 6321 . . . . . 6 (𝑑𝐷 → (𝐹𝑑) = ( lastS ‘𝑑))
10 fveq2 6229 . . . . . . 7 (𝑡 = 𝑥 → ( lastS ‘𝑡) = ( lastS ‘𝑥))
11 fvex 6239 . . . . . . 7 ( lastS ‘𝑥) ∈ V
1210, 5, 11fvmpt 6321 . . . . . 6 (𝑥𝐷 → (𝐹𝑥) = ( lastS ‘𝑥))
139, 12eqeqan12d 2667 . . . . 5 ((𝑑𝐷𝑥𝐷) → ((𝐹𝑑) = (𝐹𝑥) ↔ ( lastS ‘𝑑) = ( lastS ‘𝑥)))
1413adantl 481 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((𝐹𝑑) = (𝐹𝑥) ↔ ( lastS ‘𝑑) = ( lastS ‘𝑥)))
15 fveq2 6229 . . . . . . . . 9 (𝑤 = 𝑑 → (#‘𝑤) = (#‘𝑑))
1615eqeq1d 2653 . . . . . . . 8 (𝑤 = 𝑑 → ((#‘𝑤) = (𝑁 + 2) ↔ (#‘𝑑) = (𝑁 + 2)))
17 oveq1 6697 . . . . . . . . 9 (𝑤 = 𝑑 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑑 substr ⟨0, (𝑁 + 1)⟩))
1817eqeq1d 2653 . . . . . . . 8 (𝑤 = 𝑑 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
19 fveq2 6229 . . . . . . . . . 10 (𝑤 = 𝑑 → ( lastS ‘𝑤) = ( lastS ‘𝑑))
2019preq2d 4307 . . . . . . . . 9 (𝑤 = 𝑑 → {( lastS ‘𝑊), ( lastS ‘𝑤)} = {( lastS ‘𝑊), ( lastS ‘𝑑)})
2120eleq1d 2715 . . . . . . . 8 (𝑤 = 𝑑 → ({( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸))
2216, 18, 213anbi123d 1439 . . . . . . 7 (𝑤 = 𝑑 → (((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) ↔ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)))
2322, 3elrab2 3399 . . . . . 6 (𝑑𝐷 ↔ (𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)))
24 fveq2 6229 . . . . . . . . 9 (𝑤 = 𝑥 → (#‘𝑤) = (#‘𝑥))
2524eqeq1d 2653 . . . . . . . 8 (𝑤 = 𝑥 → ((#‘𝑤) = (𝑁 + 2) ↔ (#‘𝑥) = (𝑁 + 2)))
26 oveq1 6697 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩))
2726eqeq1d 2653 . . . . . . . 8 (𝑤 = 𝑥 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
28 fveq2 6229 . . . . . . . . . 10 (𝑤 = 𝑥 → ( lastS ‘𝑤) = ( lastS ‘𝑥))
2928preq2d 4307 . . . . . . . . 9 (𝑤 = 𝑥 → {( lastS ‘𝑊), ( lastS ‘𝑤)} = {( lastS ‘𝑊), ( lastS ‘𝑥)})
3029eleq1d 2715 . . . . . . . 8 (𝑤 = 𝑥 → ({( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))
3125, 27, 303anbi123d 1439 . . . . . . 7 (𝑤 = 𝑥 → (((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) ↔ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)))
3231, 3elrab2 3399 . . . . . 6 (𝑥𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)))
33 eqtr3 2672 . . . . . . . . . . . . . . . . 17 (((#‘𝑑) = (𝑁 + 2) ∧ (#‘𝑥) = (𝑁 + 2)) → (#‘𝑑) = (#‘𝑥))
3433expcom 450 . . . . . . . . . . . . . . . 16 ((#‘𝑥) = (𝑁 + 2) → ((#‘𝑑) = (𝑁 + 2) → (#‘𝑑) = (#‘𝑥)))
35343ad2ant1 1102 . . . . . . . . . . . . . . 15 (((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸) → ((#‘𝑑) = (𝑁 + 2) → (#‘𝑑) = (#‘𝑥)))
3635adantl 481 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)) → ((#‘𝑑) = (𝑁 + 2) → (#‘𝑑) = (#‘𝑥)))
3736com12 32 . . . . . . . . . . . . 13 ((#‘𝑑) = (𝑁 + 2) → ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)) → (#‘𝑑) = (#‘𝑥)))
38373ad2ant1 1102 . . . . . . . . . . . 12 (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸) → ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)) → (#‘𝑑) = (#‘𝑥)))
3938adantl 481 . . . . . . . . . . 11 ((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) → ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)) → (#‘𝑑) = (#‘𝑥)))
4039imp 444 . . . . . . . . . 10 (((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) → (#‘𝑑) = (#‘𝑥))
4140adantr 480 . . . . . . . . 9 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → (#‘𝑑) = (#‘𝑥))
4241adantr 480 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) → (#‘𝑑) = (#‘𝑥))
43 simpr 476 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) → ( lastS ‘𝑑) = ( lastS ‘𝑥))
44 eqtr3 2672 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩))
45 1e2m1 11174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 = (2 − 1)
4645a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 1 = (2 − 1))
4746oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0 → (𝑁 + 1) = (𝑁 + (2 − 1)))
48 nn0cn 11340 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
49 2cnd 11131 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
50 1cnd 10094 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
5148, 49, 50addsubassd 10450 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
5247, 51eqtr4d 2688 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0 → (𝑁 + 1) = ((𝑁 + 2) − 1))
5352adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ ℕ0 ∧ (#‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((𝑁 + 2) − 1))
54 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝑑) = (𝑁 + 2) → ((#‘𝑑) − 1) = ((𝑁 + 2) − 1))
5554eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝑑) = (𝑁 + 2) → ((𝑁 + 1) = ((#‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
5655adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ ℕ0 ∧ (#‘𝑑) = (𝑁 + 2)) → ((𝑁 + 1) = ((#‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
5753, 56mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ0 ∧ (#‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((#‘𝑑) − 1))
58 opeq2 4434 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑁 + 1) = ((#‘𝑑) − 1) → ⟨0, (𝑁 + 1)⟩ = ⟨0, ((#‘𝑑) − 1)⟩)
5958oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((#‘𝑑) − 1) → (𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩))
6058oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((#‘𝑑) − 1) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))
6159, 60eqeq12d 2666 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) = ((#‘𝑑) − 1) → ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ↔ (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩)))
6257, 61syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ0 ∧ (#‘𝑑) = (𝑁 + 2)) → ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ↔ (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩)))
6362biimpd 219 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ0 ∧ (#‘𝑑) = (𝑁 + 2)) → ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩)))
6463ex 449 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → ((#‘𝑑) = (𝑁 + 2) → ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
6564com13 88 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ((#‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
6644, 65syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → ((#‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
6766ex 449 . . . . . . . . . . . . . . . . . 18 ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → ((#‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩)))))
6867com23 86 . . . . . . . . . . . . . . . . 17 ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → ((#‘𝑑) = (𝑁 + 2) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩)))))
6968impcom 445 . . . . . . . . . . . . . . . 16 (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
7069com12 32 . . . . . . . . . . . . . . 15 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
71703ad2ant2 1103 . . . . . . . . . . . . . 14 (((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸) → (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
7271adantl 481 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)) → (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
7372com12 32 . . . . . . . . . . . 12 (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
74733adant3 1101 . . . . . . . . . . 11 (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸) → ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
7574adantl 481 . . . . . . . . . 10 ((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) → ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
7675imp31 447 . . . . . . . . 9 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))
7776adantr 480 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))
78 simpl 472 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) → 𝑑 ∈ Word 𝑉)
79 simpl 472 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)) → 𝑥 ∈ Word 𝑉)
8078, 79anim12i 589 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) → (𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉))
8180adantr 480 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → (𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉))
82 nn0re 11339 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
83 2re 11128 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℝ
8483a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
85 nn0ge0 11356 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
86 2pos 11150 . . . . . . . . . . . . . . . . . . . . . 22 0 < 2
8786a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 < 2)
8882, 84, 85, 87addgegt0d 10639 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
8988adantl 481 . . . . . . . . . . . . . . . . . . 19 (((#‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
90 breq2 4689 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑑) = (𝑁 + 2) → (0 < (#‘𝑑) ↔ 0 < (𝑁 + 2)))
9190adantr 480 . . . . . . . . . . . . . . . . . . 19 (((#‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (#‘𝑑) ↔ 0 < (𝑁 + 2)))
9289, 91mpbird 247 . . . . . . . . . . . . . . . . . 18 (((#‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (#‘𝑑))
93 hashgt0n0 13194 . . . . . . . . . . . . . . . . . 18 ((𝑑 ∈ Word 𝑉 ∧ 0 < (#‘𝑑)) → 𝑑 ≠ ∅)
9492, 93sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑑 ≠ ∅)
9594exp32 630 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 → ((#‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
9695com12 32 . . . . . . . . . . . . . . 15 ((#‘𝑑) = (𝑁 + 2) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
97963ad2ant1 1102 . . . . . . . . . . . . . 14 (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
9897impcom 445 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) → (𝑁 ∈ ℕ0𝑑 ≠ ∅))
9998adantr 480 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ0𝑑 ≠ ∅))
10099imp 444 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → 𝑑 ≠ ∅)
10188adantl 481 . . . . . . . . . . . . . . . . . . 19 (((#‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
102 breq2 4689 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑥) = (𝑁 + 2) → (0 < (#‘𝑥) ↔ 0 < (𝑁 + 2)))
103102adantr 480 . . . . . . . . . . . . . . . . . . 19 (((#‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (#‘𝑥) ↔ 0 < (𝑁 + 2)))
104101, 103mpbird 247 . . . . . . . . . . . . . . . . . 18 (((#‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (#‘𝑥))
105 hashgt0n0 13194 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ Word 𝑉 ∧ 0 < (#‘𝑥)) → 𝑥 ≠ ∅)
106104, 105sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑥 ≠ ∅)
107106exp32 630 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Word 𝑉 → ((#‘𝑥) = (𝑁 + 2) → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
108107com12 32 . . . . . . . . . . . . . . 15 ((#‘𝑥) = (𝑁 + 2) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
1091083ad2ant1 1102 . . . . . . . . . . . . . 14 (((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
110109impcom 445 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0𝑥 ≠ ∅))
111110adantl 481 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ0𝑥 ≠ ∅))
112111imp 444 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → 𝑥 ≠ ∅)
11381, 100, 112jca32 557 . . . . . . . . . 10 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
114113adantr 480 . . . . . . . . 9 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
115 simpl 472 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 𝑑 ∈ Word 𝑉)
116115adantr 480 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑑 ∈ Word 𝑉)
117 simpr 476 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 𝑥 ∈ Word 𝑉)
118117adantr 480 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ Word 𝑉)
119 hashneq0 13193 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 → (0 < (#‘𝑑) ↔ 𝑑 ≠ ∅))
120119biimprd 238 . . . . . . . . . . . . . . 15 (𝑑 ∈ Word 𝑉 → (𝑑 ≠ ∅ → 0 < (#‘𝑑)))
121120adantr 480 . . . . . . . . . . . . . 14 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → (𝑑 ≠ ∅ → 0 < (#‘𝑑)))
122121com12 32 . . . . . . . . . . . . 13 (𝑑 ≠ ∅ → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 0 < (#‘𝑑)))
123122adantr 480 . . . . . . . . . . . 12 ((𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅) → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 0 < (#‘𝑑)))
124123impcom 445 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 0 < (#‘𝑑))
125 2swrd1eqwrdeq 13500 . . . . . . . . . . 11 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉 ∧ 0 < (#‘𝑑)) → (𝑑 = 𝑥 ↔ ((#‘𝑑) = (#‘𝑥) ∧ ((𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)))))
126116, 118, 124, 125syl3anc 1366 . . . . . . . . . 10 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → (𝑑 = 𝑥 ↔ ((#‘𝑑) = (#‘𝑥) ∧ ((𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)))))
127 ancom 465 . . . . . . . . . . . 12 (((𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) ↔ (( lastS ‘𝑑) = ( lastS ‘𝑥) ∧ (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩)))
128127anbi2i 730 . . . . . . . . . . 11 (((#‘𝑑) = (#‘𝑥) ∧ ((𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥))) ↔ ((#‘𝑑) = (#‘𝑥) ∧ (( lastS ‘𝑑) = ( lastS ‘𝑥) ∧ (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
129 3anass 1059 . . . . . . . . . . 11 (((#‘𝑑) = (#‘𝑥) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥) ∧ (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩)) ↔ ((#‘𝑑) = (#‘𝑥) ∧ (( lastS ‘𝑑) = ( lastS ‘𝑥) ∧ (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
130128, 129bitr4i 267 . . . . . . . . . 10 (((#‘𝑑) = (#‘𝑥) ∧ ((𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥))) ↔ ((#‘𝑑) = (#‘𝑥) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥) ∧ (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩)))
131126, 130syl6bb 276 . . . . . . . . 9 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → (𝑑 = 𝑥 ↔ ((#‘𝑑) = (#‘𝑥) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥) ∧ (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
132114, 131syl 17 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) → (𝑑 = 𝑥 ↔ ((#‘𝑑) = (#‘𝑥) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥) ∧ (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
13342, 43, 77, 132mpbir3and 1264 . . . . . . 7 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) → 𝑑 = 𝑥)
134133exp31 629 . . . . . 6 (((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ0 → (( lastS ‘𝑑) = ( lastS ‘𝑥) → 𝑑 = 𝑥)))
13523, 32, 134syl2anb 495 . . . . 5 ((𝑑𝐷𝑥𝐷) → (𝑁 ∈ ℕ0 → (( lastS ‘𝑑) = ( lastS ‘𝑥) → 𝑑 = 𝑥)))
136135impcom 445 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → (( lastS ‘𝑑) = ( lastS ‘𝑥) → 𝑑 = 𝑥))
13714, 136sylbid 230 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥))
138137ralrimivva 3000 . 2 (𝑁 ∈ ℕ0 → ∀𝑑𝐷𝑥𝐷 ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥))
139 dff13 6552 . 2 (𝐹:𝐷1-1𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑑𝐷𝑥𝐷 ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥)))
1406, 138, 139sylanbrc 699 1 (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  {crab 2945  ∅c0 3948  {cpr 4212  ⟨cop 4216   class class class wbr 4685   ↦ cmpt 4762  ⟶wf 5922  –1-1→wf1 5923  ‘cfv 5926  (class class class)co 6690  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112   − cmin 10304  2c2 11108  ℕ0cn0 11330  #chash 13157  Word cword 13323   lastS clsw 13324   substr csubstr 13327  Vtxcvtx 25919  Edgcedg 25984 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-lsw 13332  df-s1 13334  df-substr 13335 This theorem is referenced by:  wwlksnextbij0  26864
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