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Theorem wwlkextinj 26258
 Description: Lemma 2 for wwlkextbij 26261. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
Hypotheses
Ref Expression
wwlkextbij.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}
wwlkextbij.r 𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}
wwlkextbij.f 𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))
Assertion
Ref Expression
wwlkextinj (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)
Distinct variable groups:   𝑡,𝐷   𝑛,𝐸,𝑤   𝑡,𝑁,𝑤   𝑡,𝑅   𝑛,𝑉,𝑡,𝑤   𝑛,𝑊,𝑡,𝑤
Allowed substitution hints:   𝐷(𝑤,𝑛)   𝑅(𝑤,𝑛)   𝐸(𝑡)   𝐹(𝑤,𝑡,𝑛)   𝑁(𝑛)

Proof of Theorem wwlkextinj
Dummy variables 𝑑 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlkextbij.d . . 3 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}
2 wwlkextbij.r . . 3 𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}
3 wwlkextbij.f . . 3 𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))
41, 2, 3wwlkextfun 26257 . 2 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
5 fveq2 6103 . . . . . . 7 (𝑡 = 𝑑 → ( lastS ‘𝑡) = ( lastS ‘𝑑))
6 fvex 6113 . . . . . . 7 ( lastS ‘𝑑) ∈ V
75, 3, 6fvmpt 6191 . . . . . 6 (𝑑𝐷 → (𝐹𝑑) = ( lastS ‘𝑑))
8 fveq2 6103 . . . . . . 7 (𝑡 = 𝑥 → ( lastS ‘𝑡) = ( lastS ‘𝑥))
9 fvex 6113 . . . . . . 7 ( lastS ‘𝑥) ∈ V
108, 3, 9fvmpt 6191 . . . . . 6 (𝑥𝐷 → (𝐹𝑥) = ( lastS ‘𝑥))
117, 10eqeqan12d 2626 . . . . 5 ((𝑑𝐷𝑥𝐷) → ((𝐹𝑑) = (𝐹𝑥) ↔ ( lastS ‘𝑑) = ( lastS ‘𝑥)))
1211adantl 481 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((𝐹𝑑) = (𝐹𝑥) ↔ ( lastS ‘𝑑) = ( lastS ‘𝑥)))
13 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑑 → (#‘𝑤) = (#‘𝑑))
1413eqeq1d 2612 . . . . . . . 8 (𝑤 = 𝑑 → ((#‘𝑤) = (𝑁 + 2) ↔ (#‘𝑑) = (𝑁 + 2)))
15 oveq1 6556 . . . . . . . . 9 (𝑤 = 𝑑 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑑 substr ⟨0, (𝑁 + 1)⟩))
1615eqeq1d 2612 . . . . . . . 8 (𝑤 = 𝑑 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
17 fveq2 6103 . . . . . . . . . 10 (𝑤 = 𝑑 → ( lastS ‘𝑤) = ( lastS ‘𝑑))
1817preq2d 4219 . . . . . . . . 9 (𝑤 = 𝑑 → {( lastS ‘𝑊), ( lastS ‘𝑤)} = {( lastS ‘𝑊), ( lastS ‘𝑑)})
1918eleq1d 2672 . . . . . . . 8 (𝑤 = 𝑑 → ({( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸))
2014, 16, 193anbi123d 1391 . . . . . . 7 (𝑤 = 𝑑 → (((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) ↔ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)))
2120, 1elrab2 3333 . . . . . 6 (𝑑𝐷 ↔ (𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)))
22 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑥 → (#‘𝑤) = (#‘𝑥))
2322eqeq1d 2612 . . . . . . . 8 (𝑤 = 𝑥 → ((#‘𝑤) = (𝑁 + 2) ↔ (#‘𝑥) = (𝑁 + 2)))
24 oveq1 6556 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩))
2524eqeq1d 2612 . . . . . . . 8 (𝑤 = 𝑥 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
26 fveq2 6103 . . . . . . . . . 10 (𝑤 = 𝑥 → ( lastS ‘𝑤) = ( lastS ‘𝑥))
2726preq2d 4219 . . . . . . . . 9 (𝑤 = 𝑥 → {( lastS ‘𝑊), ( lastS ‘𝑤)} = {( lastS ‘𝑊), ( lastS ‘𝑥)})
2827eleq1d 2672 . . . . . . . 8 (𝑤 = 𝑥 → ({( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))
2923, 25, 283anbi123d 1391 . . . . . . 7 (𝑤 = 𝑥 → (((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) ↔ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸)))
3029, 1elrab2 3333 . . . . . 6 (𝑥𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸)))
31 eqtr3 2631 . . . . . . . . . . . . . . . . 17 (((#‘𝑑) = (𝑁 + 2) ∧ (#‘𝑥) = (𝑁 + 2)) → (#‘𝑑) = (#‘𝑥))
3231expcom 450 . . . . . . . . . . . . . . . 16 ((#‘𝑥) = (𝑁 + 2) → ((#‘𝑑) = (𝑁 + 2) → (#‘𝑑) = (#‘𝑥)))
33323ad2ant1 1075 . . . . . . . . . . . . . . 15 (((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸) → ((#‘𝑑) = (𝑁 + 2) → (#‘𝑑) = (#‘𝑥)))
3433adantl 481 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸)) → ((#‘𝑑) = (𝑁 + 2) → (#‘𝑑) = (#‘𝑥)))
3534com12 32 . . . . . . . . . . . . 13 ((#‘𝑑) = (𝑁 + 2) → ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸)) → (#‘𝑑) = (#‘𝑥)))
36353ad2ant1 1075 . . . . . . . . . . . 12 (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸) → ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸)) → (#‘𝑑) = (#‘𝑥)))
3736adantl 481 . . . . . . . . . . 11 ((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) → ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸)) → (#‘𝑑) = (#‘𝑥)))
3837imp 444 . . . . . . . . . 10 (((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) → (#‘𝑑) = (#‘𝑥))
3938adantr 480 . . . . . . . . 9 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) ∧ 𝑁 ∈ ℕ0) → (#‘𝑑) = (#‘𝑥))
4039adantr 480 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) → (#‘𝑑) = (#‘𝑥))
41 simpr 476 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) → ( lastS ‘𝑑) = ( lastS ‘𝑥))
42 eqtr3 2631 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩))
43 1e2m1 11013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 = (2 − 1)
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 1 = (2 − 1))
4544oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0 → (𝑁 + 1) = (𝑁 + (2 − 1)))
46 nn0cn 11179 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
47 2cnd 10970 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
48 1cnd 9935 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
4946, 47, 48addsubassd 10291 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
5045, 49eqtr4d 2647 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0 → (𝑁 + 1) = ((𝑁 + 2) − 1))
5150adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ ℕ0 ∧ (#‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((𝑁 + 2) − 1))
52 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝑑) = (𝑁 + 2) → ((#‘𝑑) − 1) = ((𝑁 + 2) − 1))
5352eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝑑) = (𝑁 + 2) → ((𝑁 + 1) = ((#‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
5453adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ ℕ0 ∧ (#‘𝑑) = (𝑁 + 2)) → ((𝑁 + 1) = ((#‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
5551, 54mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ0 ∧ (#‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((#‘𝑑) − 1))
56 opeq2 4341 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑁 + 1) = ((#‘𝑑) − 1) → ⟨0, (𝑁 + 1)⟩ = ⟨0, ((#‘𝑑) − 1)⟩)
5756oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((#‘𝑑) − 1) → (𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩))
5856oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((#‘𝑑) − 1) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))
5957, 58eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
6055, 59syl 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)⟩)))
6160biimpd 218 . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
6261ex 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)⟩))))
6362com13 86 . . . . . . . . . . . . . . . . . . . 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)⟩))))
6442, 63syl 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)⟩))))
6564ex 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)⟩)))))
6665com23 84 . . . . . . . . . . . . . . . . 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)⟩)))))
6766impcom 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)⟩))))
6867com12 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)⟩))))
69683ad2ant2 1076 . . . . . . . . . . . . . 14 (((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸) → (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
7069adantl 481 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸)) → (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
7170com12 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)} ∈ ran 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
72713adant3 1074 . . . . . . . . . . 11 (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸) → ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
7372adantl 481 . . . . . . . . . 10 ((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) → ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))))
7473imp31 447 . . . . . . . . 9 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) ∧ 𝑁 ∈ ℕ0) → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))
7574adantr 480 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) → (𝑑 substr ⟨0, ((#‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((#‘𝑑) − 1)⟩))
76 simpl 472 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) → 𝑑 ∈ Word 𝑉)
77 simpl 472 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸)) → 𝑥 ∈ Word 𝑉)
7876, 77anim12i 588 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) → (𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉))
7978adantr 480 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) ∧ 𝑁 ∈ ℕ0) → (𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉))
80 nn0re 11178 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
81 2re 10967 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℝ
8281a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
83 nn0ge0 11195 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
84 2pos 10989 . . . . . . . . . . . . . . . . . . . . . 22 0 < 2
8584a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 < 2)
8680, 82, 83, 85addgegt0d 10480 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
8786adantl 481 . . . . . . . . . . . . . . . . . . 19 (((#‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
88 breq2 4587 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑑) = (𝑁 + 2) → (0 < (#‘𝑑) ↔ 0 < (𝑁 + 2)))
8988adantr 480 . . . . . . . . . . . . . . . . . . 19 (((#‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (#‘𝑑) ↔ 0 < (𝑁 + 2)))
9087, 89mpbird 246 . . . . . . . . . . . . . . . . . 18 (((#‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (#‘𝑑))
91 hashgt0n0 13017 . . . . . . . . . . . . . . . . . 18 ((𝑑 ∈ Word 𝑉 ∧ 0 < (#‘𝑑)) → 𝑑 ≠ ∅)
9290, 91sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑑 ≠ ∅)
9392exp32 629 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 → ((#‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
9493com12 32 . . . . . . . . . . . . . . 15 ((#‘𝑑) = (𝑁 + 2) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
95943ad2ant1 1075 . . . . . . . . . . . . . 14 (((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
9695impcom 445 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) → (𝑁 ∈ ℕ0𝑑 ≠ ∅))
9796adantr 480 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) → (𝑁 ∈ ℕ0𝑑 ≠ ∅))
9897imp 444 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) ∧ 𝑁 ∈ ℕ0) → 𝑑 ≠ ∅)
9986adantl 481 . . . . . . . . . . . . . . . . . . 19 (((#‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
100 breq2 4587 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑥) = (𝑁 + 2) → (0 < (#‘𝑥) ↔ 0 < (𝑁 + 2)))
101100adantr 480 . . . . . . . . . . . . . . . . . . 19 (((#‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (#‘𝑥) ↔ 0 < (𝑁 + 2)))
10299, 101mpbird 246 . . . . . . . . . . . . . . . . . 18 (((#‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (#‘𝑥))
103 hashgt0n0 13017 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ Word 𝑉 ∧ 0 < (#‘𝑥)) → 𝑥 ≠ ∅)
104102, 103sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑥 ≠ ∅)
105104exp32 629 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Word 𝑉 → ((#‘𝑥) = (𝑁 + 2) → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
106105com12 32 . . . . . . . . . . . . . . 15 ((#‘𝑥) = (𝑁 + 2) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
1071063ad2ant1 1075 . . . . . . . . . . . . . 14 (((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
108107impcom 445 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸)) → (𝑁 ∈ ℕ0𝑥 ≠ ∅))
109108adantl 481 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) → (𝑁 ∈ ℕ0𝑥 ≠ ∅))
110109imp 444 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) ∧ 𝑁 ∈ ℕ0) → 𝑥 ≠ ∅)
11179, 98, 110jca32 556 . . . . . . . . . 10 ((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) ∧ 𝑁 ∈ ℕ0) → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
112111adantr 480 . . . . . . . . 9 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
113 simpl 472 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 𝑑 ∈ Word 𝑉)
114113adantr 480 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑑 ∈ Word 𝑉)
115 simpr 476 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 𝑥 ∈ Word 𝑉)
116115adantr 480 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ Word 𝑉)
117 hashneq0 13016 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 → (0 < (#‘𝑑) ↔ 𝑑 ≠ ∅))
118117biimprd 237 . . . . . . . . . . . . . . 15 (𝑑 ∈ Word 𝑉 → (𝑑 ≠ ∅ → 0 < (#‘𝑑)))
119118adantr 480 . . . . . . . . . . . . . 14 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → (𝑑 ≠ ∅ → 0 < (#‘𝑑)))
120119com12 32 . . . . . . . . . . . . 13 (𝑑 ≠ ∅ → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 0 < (#‘𝑑)))
121120adantr 480 . . . . . . . . . . . 12 ((𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅) → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 0 < (#‘𝑑)))
122121impcom 445 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 0 < (#‘𝑑))
123 2swrd1eqwrdeq 13306 . . . . . . . . . . 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)))))
124114, 116, 122, 123syl3anc 1318 . . . . . . . . . 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)))))
125 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)⟩)))
126125anbi2i 726 . . . . . . . . . . 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)⟩))))
127 3anass 1035 . . . . . . . . . . 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)⟩))))
128126, 127bitr4i 266 . . . . . . . . . 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)⟩)))
129124, 128syl6bb 275 . . . . . . . . 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)⟩))))
130112, 129syl 17 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 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)⟩))))
13140, 41, 75, 130mpbir3and 1238 . . . . . . 7 (((((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ ( lastS ‘𝑑) = ( lastS ‘𝑥)) → 𝑑 = 𝑥)
132131exp31 628 . . . . . 6 (((𝑑 ∈ Word 𝑉 ∧ ((#‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ ran 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((#‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑥)} ∈ ran 𝐸))) → (𝑁 ∈ ℕ0 → (( lastS ‘𝑑) = ( lastS ‘𝑥) → 𝑑 = 𝑥)))
13321, 30, 132syl2anb 495 . . . . 5 ((𝑑𝐷𝑥𝐷) → (𝑁 ∈ ℕ0 → (( lastS ‘𝑑) = ( lastS ‘𝑥) → 𝑑 = 𝑥)))
134133impcom 445 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → (( lastS ‘𝑑) = ( lastS ‘𝑥) → 𝑑 = 𝑥))
13512, 134sylbid 229 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥))
136135ralrimivva 2954 . 2 (𝑁 ∈ ℕ0 → ∀𝑑𝐷𝑥𝐷 ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥))
137 dff13 6416 . 2 (𝐹:𝐷1-1𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑑𝐷𝑥𝐷 ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥)))
1384, 136, 137sylanbrc 695 1 (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  ∅c0 3874  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  2c2 10947  ℕ0cn0 11169  #chash 12979  Word cword 13146   lastS clsw 13147   substr csubstr 13150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-s1 13157  df-substr 13158 This theorem is referenced by:  wwlkextbij0  26260
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