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Theorem wwlksnextsurj 27221
Description: Lemma for wwlksnextbij 27228. (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
wwlksnextsurj (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 wwlksnextsurj
Dummy variables 𝑖 𝑑 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlksnextbij0.v . . . 4 𝑉 = (Vtx‘𝐺)
21wwlknbp 27148 . . 3 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
3 simp2 1171 . . 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 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
6 wwlksnextbij0.r . . . 4 𝑅 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
7 wwlksnextbij0.f . . . 4 𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))
81, 4, 5, 6, 7wwlksnextfun 27219 . . 3 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
92, 3, 83syl 18 . 2 (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷𝑅)
10 preq2 4489 . . . . . 6 (𝑛 = 𝑟 → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), 𝑟})
1110eleq1d 2891 . . . . 5 (𝑛 = 𝑟 → ({(lastS‘𝑊), 𝑛} ∈ 𝐸 ↔ {(lastS‘𝑊), 𝑟} ∈ 𝐸))
1211, 6elrab2 3589 . . . 4 (𝑟𝑅 ↔ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸))
131, 4wwlksnext 27211 . . . . . . . . . . 11 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
14133expb 1153 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
15 s1cl 13669 . . . . . . . . . . . . . . . . . 18 (𝑟𝑉 → ⟨“𝑟”⟩ ∈ Word 𝑉)
16 pfxccat1 13788 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑟”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊)
1715, 16sylan2 586 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ Word 𝑉𝑟𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊)
1817ex 403 . . . . . . . . . . . . . . . 16 (𝑊 ∈ Word 𝑉 → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
1918adantr 474 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
20 oveq2 6918 . . . . . . . . . . . . . . . . . 18 ((𝑁 + 1) = (♯‘𝑊) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)))
2120eqcoms 2833 . . . . . . . . . . . . . . . . 17 ((♯‘𝑊) = (𝑁 + 1) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)))
2221eqeq1d 2827 . . . . . . . . . . . . . . . 16 ((♯‘𝑊) = (𝑁 + 1) → (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
2322adantl 475 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
2419, 23sylibrd 251 . . . . . . . . . . . . . 14 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
25243adant3 1166 . . . . . . . . . . . . 13 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
261, 4wwlknp 27149 . . . . . . . . . . . . 13 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
2725, 26syl11 33 . . . . . . . . . . . 12 (𝑟𝑉 → (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
2827adantr 474 . . . . . . . . . . 11 ((𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
2928impcom 398 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊)
30 lswccats1 13701 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Word 𝑉𝑟𝑉) → (lastS‘(𝑊 ++ ⟨“𝑟”⟩)) = 𝑟)
3130eqcomd 2831 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉𝑟𝑉) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
3231ex 403 . . . . . . . . . . . . . . . . 17 (𝑊 ∈ Word 𝑉 → (𝑟𝑉𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
33323ad2ant3 1169 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑟𝑉𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
342, 33syl 17 . . . . . . . . . . . . . . 15 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑟𝑉𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
3534imp 397 . . . . . . . . . . . . . 14 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑉) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
3635preq2d 4495 . . . . . . . . . . . . 13 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑉) → {(lastS‘𝑊), 𝑟} = {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))})
3736eleq1d 2891 . . . . . . . . . . . 12 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑉) → ({(lastS‘𝑊), 𝑟} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
3837biimpd 221 . . . . . . . . . . 11 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑉) → ({(lastS‘𝑊), 𝑟} ∈ 𝐸 → {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
3938impr 448 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)
4014, 29, 39jca32 511 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)))
4133, 2syl11 33 . . . . . . . . . . 11 (𝑟𝑉 → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
4241adantr 474 . . . . . . . . . 10 ((𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
4342impcom 398 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
44 ovexd 6944 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ V)
45 eleq1 2894 . . . . . . . . . . . . . . 15 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
46 oveq1 6917 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)))
4746eqeq1d 2827 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
48 fveq2 6437 . . . . . . . . . . . . . . . . . 18 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (lastS‘𝑑) = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
4948preq2d 4495 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → {(lastS‘𝑊), (lastS‘𝑑)} = {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))})
5049eleq1d 2891 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ({(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
5147, 50anbi12d 624 . . . . . . . . . . . . . . 15 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) ↔ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)))
5245, 51anbi12d 624 . . . . . . . . . . . . . 14 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ↔ ((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))))
5348eqeq2d 2835 . . . . . . . . . . . . . 14 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑟 = (lastS‘𝑑) ↔ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
5452, 53anbi12d 624 . . . . . . . . . . . . 13 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑)) ↔ (((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))))
5554bicomd 215 . . . . . . . . . . . 12 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑))))
5655adantl 475 . . . . . . . . . . 11 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) ∧ 𝑑 = (𝑊 ++ ⟨“𝑟”⟩)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑))))
5756biimpd 221 . . . . . . . . . 10 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) ∧ 𝑑 = (𝑊 ++ ⟨“𝑟”⟩)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))) → ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑))))
5844, 57spcimedv 3509 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))) → ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑))))
5940, 43, 58mp2and 690 . . . . . . . 8 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑)))
60 oveq1 6917 . . . . . . . . . . . . 13 (𝑤 = 𝑑 → (𝑤 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
6160eqeq1d 2827 . . . . . . . . . . . 12 (𝑤 = 𝑑 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑑 prefix (𝑁 + 1)) = 𝑊))
62 fveq2 6437 . . . . . . . . . . . . . 14 (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
6362preq2d 4495 . . . . . . . . . . . . 13 (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
6463eleq1d 2891 . . . . . . . . . . . 12 (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
6561, 64anbi12d 624 . . . . . . . . . . 11 (𝑤 = 𝑑 → (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
6665elrab 3585 . . . . . . . . . 10 (𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ↔ (𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
6766anbi1i 617 . . . . . . . . 9 ((𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = (lastS‘𝑑)) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑)))
6867exbii 1947 . . . . . . . 8 (∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = (lastS‘𝑑)) ↔ ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑)))
6959, 68sylibr 226 . . . . . . 7 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = (lastS‘𝑑)))
70 df-rex 3123 . . . . . . 7 (∃𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}𝑟 = (lastS‘𝑑) ↔ ∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = (lastS‘𝑑)))
7169, 70sylibr 226 . . . . . 6 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}𝑟 = (lastS‘𝑑))
721, 4, 5wwlksnextwrd 27218 . . . . . . . 8 (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
7372adantr 474 . . . . . . 7 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
7473rexeqdv 3357 . . . . . 6 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → (∃𝑑𝐷 𝑟 = (lastS‘𝑑) ↔ ∃𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}𝑟 = (lastS‘𝑑)))
7571, 74mpbird 249 . . . . 5 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑𝐷 𝑟 = (lastS‘𝑑))
76 fveq2 6437 . . . . . . . 8 (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
77 fvex 6450 . . . . . . . 8 (lastS‘𝑑) ∈ V
7876, 7, 77fvmpt 6533 . . . . . . 7 (𝑑𝐷 → (𝐹𝑑) = (lastS‘𝑑))
7978eqeq2d 2835 . . . . . 6 (𝑑𝐷 → (𝑟 = (𝐹𝑑) ↔ 𝑟 = (lastS‘𝑑)))
8079rexbiia 3250 . . . . 5 (∃𝑑𝐷 𝑟 = (𝐹𝑑) ↔ ∃𝑑𝐷 𝑟 = (lastS‘𝑑))
8175, 80sylibr 226 . . . 4 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑𝐷 𝑟 = (𝐹𝑑))
8212, 81sylan2b 587 . . 3 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑅) → ∃𝑑𝐷 𝑟 = (𝐹𝑑))
8382ralrimiva 3175 . 2 (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∀𝑟𝑅𝑑𝐷 𝑟 = (𝐹𝑑))
84 dffo3 6628 . 2 (𝐹:𝐷onto𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑟𝑅𝑑𝐷 𝑟 = (𝐹𝑑)))
859, 83, 84sylanbrc 578 1 (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷onto𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wex 1878  wcel 2164  wral 3117  wrex 3118  {crab 3121  Vcvv 3414  {cpr 4401  cmpt 4954  wf 6123  ontowfo 6125  cfv 6127  (class class class)co 6910  0cc0 10259  1c1 10260   + caddc 10262  2c2 11413  0cn0 11625  ..^cfzo 12767  chash 13417  Word cword 13581  lastSclsw 13629   ++ cconcat 13637  ⟨“cs1 13662   prefix cpfx 13756  Vtxcvtx 26301  Edgcedg 26352   WWalksN cwwlksn 27132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-n0 11626  df-xnn0 11698  df-z 11712  df-uz 11976  df-rp 12120  df-fz 12627  df-fzo 12768  df-hash 13418  df-word 13582  df-lsw 13630  df-concat 13638  df-s1 13663  df-substr 13708  df-pfx 13757  df-wwlks 27136  df-wwlksn 27137
This theorem is referenced by:  wwlksnextbij0  27222
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