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Theorem wwlksnextsur 41106
 Description: Lemma for wwlksnextbij 41108. (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
wwlksnextsur (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝐹:𝐷onto𝑅)
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑊   𝑡,𝐷   𝑛,𝐸,𝑤   𝑡,𝑁,𝑤   𝑡,𝑅   𝑛,𝑉,𝑤   𝑛,𝑊   𝑡,𝑛,𝑁,𝑤
Allowed substitution hints:   𝐷(𝑤,𝑛)   𝑅(𝑤,𝑛)   𝐸(𝑡)   𝐹(𝑤,𝑡,𝑛)   𝐺(𝑡,𝑛)   𝑉(𝑡)   𝑊(𝑡)

Proof of Theorem wwlksnextsur
Dummy variables 𝑖 𝑑 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlksnextbij0.v . . . 4 𝑉 = (Vtx‘𝐺)
21wwlknbp 41044 . . 3 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
3 simp2 1055 . . 3 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → 𝑁 ∈ ℕ0)
4 wwlksnextbij0.e . . . 4 𝐸 = (Edg‘𝐺)
5 wwlksnextbij0.d . . . 4 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
6 wwlksnextbij.r . . . 4 𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}
7 wwlksnextbij.f . . . 4 𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))
81, 4, 5, 6, 7wwlksnextfun 41104 . . 3 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
92, 3, 83syl 18 . 2 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝐹:𝐷𝑅)
10 preq2 4213 . . . . . 6 (𝑛 = 𝑟 → {( lastS ‘𝑊), 𝑛} = {( lastS ‘𝑊), 𝑟})
1110eleq1d 2672 . . . . 5 (𝑛 = 𝑟 → ({( lastS ‘𝑊), 𝑛} ∈ 𝐸 ↔ {( lastS ‘𝑊), 𝑟} ∈ 𝐸))
1211, 6elrab2 3333 . . . 4 (𝑟𝑅 ↔ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸))
131, 4wwlksnext 41099 . . . . . . . . . . 11 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺))
14133expb 1258 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺))
151, 4wwlknp 41045 . . . . . . . . . . . . . 14 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
16 s1cl 13235 . . . . . . . . . . . . . . . . . . 19 (𝑟𝑉 → ⟨“𝑟”⟩ ∈ Word 𝑉)
17 swrdccat1 13309 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑟”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊)
1816, 17sylan2 490 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉𝑟𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊)
1918ex 449 . . . . . . . . . . . . . . . . 17 (𝑊 ∈ Word 𝑉 → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
2019adantr 480 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
21 opeq2 4341 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 + 1) = (#‘𝑊) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑊)⟩)
2221eqcoms 2618 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑊) = (𝑁 + 1) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑊)⟩)
2322oveq2d 6565 . . . . . . . . . . . . . . . . . 18 ((#‘𝑊) = (𝑁 + 1) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩))
2423eqeq1d 2612 . . . . . . . . . . . . . . . . 17 ((#‘𝑊) = (𝑁 + 1) → (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
2524adantl 481 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
2620, 25sylibrd 248 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
27263adant3 1074 . . . . . . . . . . . . . 14 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
2815, 27syl 17 . . . . . . . . . . . . 13 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
2928com12 32 . . . . . . . . . . . 12 (𝑟𝑉 → (𝑊 ∈ (𝑁 WWalkSN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
3029adantr 480 . . . . . . . . . . 11 ((𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalkSN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
3130impcom 445 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊)
32 lswccats1 13263 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Word 𝑉𝑟𝑉) → ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)) = 𝑟)
3332eqcomd 2616 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉𝑟𝑉) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
3433ex 449 . . . . . . . . . . . . . . . . 17 (𝑊 ∈ Word 𝑉 → (𝑟𝑉𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
35343ad2ant3 1077 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑟𝑉𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
362, 35syl 17 . . . . . . . . . . . . . . 15 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑟𝑉𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
3736imp 444 . . . . . . . . . . . . . 14 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟𝑉) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
3837preq2d 4219 . . . . . . . . . . . . 13 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟𝑉) → {( lastS ‘𝑊), 𝑟} = {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))})
3938eleq1d 2672 . . . . . . . . . . . 12 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟𝑉) → ({( lastS ‘𝑊), 𝑟} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
4039biimpd 218 . . . . . . . . . . 11 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟𝑉) → ({( lastS ‘𝑊), 𝑟} ∈ 𝐸 → {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
4140impr 647 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)
4214, 31, 41jca32 556 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)))
4336com12 32 . . . . . . . . . . 11 (𝑟𝑉 → (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
4443adantr 480 . . . . . . . . . 10 ((𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
4544impcom 445 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
46 ovex 6577 . . . . . . . . . . 11 (𝑊 ++ ⟨“𝑟”⟩) ∈ V
4746a1i 11 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ V)
48 eleq1 2676 . . . . . . . . . . . . . . 15 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ↔ (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺)))
49 oveq1 6556 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 substr ⟨0, (𝑁 + 1)⟩) = ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩))
5049eqeq1d 2612 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
51 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ( lastS ‘𝑑) = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
5251preq2d 4219 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → {( lastS ‘𝑊), ( lastS ‘𝑑)} = {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))})
5352eleq1d 2672 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ({( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
5450, 53anbi12d 743 . . . . . . . . . . . . . . 15 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸) ↔ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)))
5548, 54anbi12d 743 . . . . . . . . . . . . . 14 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ↔ ((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))))
5651eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑟 = ( lastS ‘𝑑) ↔ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
5755, 56anbi12d 743 . . . . . . . . . . . . 13 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (((𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑)) ↔ (((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))))
5857bicomd 212 . . . . . . . . . . . 12 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑))))
5958adantl 481 . . . . . . . . . . 11 (((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) ∧ 𝑑 = (𝑊 ++ ⟨“𝑟”⟩)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑))))
6059biimpd 218 . . . . . . . . . 10 (((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) ∧ 𝑑 = (𝑊 ++ ⟨“𝑟”⟩)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))) → ((𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑))))
6147, 60spcimedv 3265 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))) → ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑))))
6242, 45, 61mp2and 711 . . . . . . . 8 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑)))
63 oveq1 6556 . . . . . . . . . . . . 13 (𝑤 = 𝑑 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑑 substr ⟨0, (𝑁 + 1)⟩))
6463eqeq1d 2612 . . . . . . . . . . . 12 (𝑤 = 𝑑 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
65 fveq2 6103 . . . . . . . . . . . . . 14 (𝑤 = 𝑑 → ( lastS ‘𝑤) = ( lastS ‘𝑑))
6665preq2d 4219 . . . . . . . . . . . . 13 (𝑤 = 𝑑 → {( lastS ‘𝑊), ( lastS ‘𝑤)} = {( lastS ‘𝑊), ( lastS ‘𝑑)})
6766eleq1d 2672 . . . . . . . . . . . 12 (𝑤 = 𝑑 → ({( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸))
6864, 67anbi12d 743 . . . . . . . . . . 11 (𝑤 = 𝑑 → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) ↔ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)))
6968elrab 3331 . . . . . . . . . 10 (𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} ↔ (𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)))
7069anbi1i 727 . . . . . . . . 9 ((𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = ( lastS ‘𝑑)) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑)))
7170exbii 1764 . . . . . . . 8 (∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = ( lastS ‘𝑑)) ↔ ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑)))
7262, 71sylibr 223 . . . . . . 7 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = ( lastS ‘𝑑)))
73 df-rex 2902 . . . . . . 7 (∃𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}𝑟 = ( lastS ‘𝑑) ↔ ∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = ( lastS ‘𝑑)))
7472, 73sylibr 223 . . . . . 6 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}𝑟 = ( lastS ‘𝑑))
751, 4, 5wwlksnextwrd 41103 . . . . . . . 8 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})
7675adantr 480 . . . . . . 7 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})
7776rexeqdv 3122 . . . . . 6 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → (∃𝑑𝐷 𝑟 = ( lastS ‘𝑑) ↔ ∃𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}𝑟 = ( lastS ‘𝑑)))
7874, 77mpbird 246 . . . . 5 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑𝐷 𝑟 = ( lastS ‘𝑑))
79 fveq2 6103 . . . . . . . 8 (𝑡 = 𝑑 → ( lastS ‘𝑡) = ( lastS ‘𝑑))
80 fvex 6113 . . . . . . . 8 ( lastS ‘𝑑) ∈ V
8179, 7, 80fvmpt 6191 . . . . . . 7 (𝑑𝐷 → (𝐹𝑑) = ( lastS ‘𝑑))
8281eqeq2d 2620 . . . . . 6 (𝑑𝐷 → (𝑟 = (𝐹𝑑) ↔ 𝑟 = ( lastS ‘𝑑)))
8382rexbiia 3022 . . . . 5 (∃𝑑𝐷 𝑟 = (𝐹𝑑) ↔ ∃𝑑𝐷 𝑟 = ( lastS ‘𝑑))
8478, 83sylibr 223 . . . 4 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑𝐷 𝑟 = (𝐹𝑑))
8512, 84sylan2b 491 . . 3 ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟𝑅) → ∃𝑑𝐷 𝑟 = (𝐹𝑑))
8685ralrimiva 2949 . 2 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → ∀𝑟𝑅𝑑𝐷 𝑟 = (𝐹𝑑))
87 dffo3 6282 . 2 (𝐹:𝐷onto𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑟𝑅𝑑𝐷 𝑟 = (𝐹𝑑)))
889, 86, 87sylanbrc 695 1 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝐹:𝐷onto𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  {cpr 4127  ⟨cop 4131   ↦ cmpt 4643  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150  Vtxcvtx 25673  Edgcedga 25792   WWalkSN cwwlksn 41029 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-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-map 7746  df-pm 7747  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-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-wwlks 41033  df-wwlksn 41034 This theorem is referenced by:  wwlksnextbij0  41107
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