MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wwlksnextsurj Structured version   Visualization version   GIF version

Theorem wwlksnextsurj 28252
Description: Lemma for wwlksnextbij 28254. (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 28194 . . 3 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
3 simp2 1136 . . 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 28250 . . 3 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
92, 3, 83syl 18 . 2 (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷𝑅)
10 preq2 4672 . . . . . 6 (𝑛 = 𝑟 → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), 𝑟})
1110eleq1d 2823 . . . . 5 (𝑛 = 𝑟 → ({(lastS‘𝑊), 𝑛} ∈ 𝐸 ↔ {(lastS‘𝑊), 𝑟} ∈ 𝐸))
1211, 6elrab2 3628 . . . 4 (𝑟𝑅 ↔ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸))
131, 4wwlksnext 28245 . . . . . . . . . . 11 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
14133expb 1119 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
15 s1cl 14296 . . . . . . . . . . . . . . . . . 18 (𝑟𝑉 → ⟨“𝑟”⟩ ∈ Word 𝑉)
16 pfxccat1 14404 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑟”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊)
1715, 16sylan2 593 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ Word 𝑉𝑟𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊)
1817ex 413 . . . . . . . . . . . . . . . 16 (𝑊 ∈ Word 𝑉 → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
1918adantr 481 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
20 oveq2 7277 . . . . . . . . . . . . . . . . . 18 ((𝑁 + 1) = (♯‘𝑊) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)))
2120eqcoms 2746 . . . . . . . . . . . . . . . . 17 ((♯‘𝑊) = (𝑁 + 1) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)))
2221eqeq1d 2740 . . . . . . . . . . . . . . . 16 ((♯‘𝑊) = (𝑁 + 1) → (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
2322adantl 482 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
2419, 23sylibrd 258 . . . . . . . . . . . . . 14 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
25243adant3 1131 . . . . . . . . . . . . 13 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑟𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
261, 4wwlknp 28195 . . . . . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
2928impcom 408 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊)
30 lswccats1 14333 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Word 𝑉𝑟𝑉) → (lastS‘(𝑊 ++ ⟨“𝑟”⟩)) = 𝑟)
3130eqcomd 2744 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉𝑟𝑉) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
3231ex 413 . . . . . . . . . . . . . . . . 17 (𝑊 ∈ Word 𝑉 → (𝑟𝑉𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
33323ad2ant3 1134 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑟𝑉𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
342, 33syl 17 . . . . . . . . . . . . . . 15 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑟𝑉𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
3534imp 407 . . . . . . . . . . . . . 14 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑉) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
3635preq2d 4678 . . . . . . . . . . . . 13 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑉) → {(lastS‘𝑊), 𝑟} = {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))})
3736eleq1d 2823 . . . . . . . . . . . 12 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑉) → ({(lastS‘𝑊), 𝑟} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
3837biimpd 228 . . . . . . . . . . 11 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑉) → ({(lastS‘𝑊), 𝑟} ∈ 𝐸 → {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
3938impr 455 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)
4014, 29, 39jca32 516 . . . . . . . . 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 481 . . . . . . . . . 10 ((𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
4342impcom 408 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
44 ovexd 7304 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ V)
45 eleq1 2826 . . . . . . . . . . . . . . 15 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
46 oveq1 7276 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)))
4746eqeq1d 2740 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
48 fveq2 6768 . . . . . . . . . . . . . . . . . 18 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (lastS‘𝑑) = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
4948preq2d 4678 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → {(lastS‘𝑊), (lastS‘𝑑)} = {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))})
5049eleq1d 2823 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ({(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
5147, 50anbi12d 631 . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . 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 2749 . . . . . . . . . . . . . 14 (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑟 = (lastS‘𝑑) ↔ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
5452, 53anbi12d 631 . . . . . . . . . . . . 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 222 . . . . . . . . . . . 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 482 . . . . . . . . . . 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 228 . . . . . . . . . 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 3533 . . . . . . . . 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 696 . . . . . . . 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 7276 . . . . . . . . . . . . 13 (𝑤 = 𝑑 → (𝑤 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
6160eqeq1d 2740 . . . . . . . . . . . 12 (𝑤 = 𝑑 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑑 prefix (𝑁 + 1)) = 𝑊))
62 fveq2 6768 . . . . . . . . . . . . . 14 (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
6362preq2d 4678 . . . . . . . . . . . . 13 (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
6463eleq1d 2823 . . . . . . . . . . . 12 (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
6561, 64anbi12d 631 . . . . . . . . . . 11 (𝑤 = 𝑑 → (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
6665elrab 3625 . . . . . . . . . 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 624 . . . . . . . . 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 1850 . . . . . . . 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 233 . . . . . . 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 3070 . . . . . . 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 233 . . . . . 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 28249 . . . . . . . 8 (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
7372adantr 481 . . . . . . 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 3348 . . . . . 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 256 . . . . 5 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑𝐷 𝑟 = (lastS‘𝑑))
76 fveq2 6768 . . . . . . . 8 (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
77 fvex 6781 . . . . . . . 8 (lastS‘𝑑) ∈ V
7876, 7, 77fvmpt 6869 . . . . . . 7 (𝑑𝐷 → (𝐹𝑑) = (lastS‘𝑑))
7978eqeq2d 2749 . . . . . 6 (𝑑𝐷 → (𝑟 = (𝐹𝑑) ↔ 𝑟 = (lastS‘𝑑)))
8079rexbiia 3179 . . . . 5 (∃𝑑𝐷 𝑟 = (𝐹𝑑) ↔ ∃𝑑𝐷 𝑟 = (lastS‘𝑑))
8175, 80sylibr 233 . . . 4 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑𝐷 𝑟 = (𝐹𝑑))
8212, 81sylan2b 594 . . 3 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟𝑅) → ∃𝑑𝐷 𝑟 = (𝐹𝑑))
8382ralrimiva 3103 . 2 (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∀𝑟𝑅𝑑𝐷 𝑟 = (𝐹𝑑))
84 dffo3 6972 . 2 (𝐹:𝐷onto𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑟𝑅𝑑𝐷 𝑟 = (𝐹𝑑)))
859, 83, 84sylanbrc 583 1 (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷onto𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  wral 3064  wrex 3065  {crab 3068  Vcvv 3431  {cpr 4565  cmpt 5158  wf 6424  ontowfo 6426  cfv 6428  (class class class)co 7269  0cc0 10860  1c1 10861   + caddc 10863  2c2 12017  0cn0 12222  ..^cfzo 13371  chash 14033  Word cword 14206  lastSclsw 14254   ++ cconcat 14262  ⟨“cs1 14289   prefix cpfx 14372  Vtxcvtx 27355  Edgcedg 27406   WWalksN cwwlksn 28178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5210  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7580  ax-cnex 10916  ax-resscn 10917  ax-1cn 10918  ax-icn 10919  ax-addcl 10920  ax-addrcl 10921  ax-mulcl 10922  ax-mulrcl 10923  ax-mulcom 10924  ax-addass 10925  ax-mulass 10926  ax-distr 10927  ax-i2m1 10928  ax-1ne0 10929  ax-1rid 10930  ax-rnegex 10931  ax-rrecex 10932  ax-cnre 10933  ax-pre-lttri 10934  ax-pre-lttrn 10935  ax-pre-ltadd 10936  ax-pre-mulgt0 10937
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-int 4882  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5486  df-eprel 5492  df-po 5500  df-so 5501  df-fr 5541  df-we 5543  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-pred 6197  df-ord 6264  df-on 6265  df-lim 6266  df-suc 6267  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fo 6434  df-f1o 6435  df-fv 6436  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7705  df-1st 7822  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-1o 8286  df-er 8487  df-map 8606  df-en 8723  df-dom 8724  df-sdom 8725  df-fin 8726  df-card 9686  df-pnf 11000  df-mnf 11001  df-xr 11002  df-ltxr 11003  df-le 11004  df-sub 11196  df-neg 11197  df-nn 11963  df-2 12025  df-n0 12223  df-xnn0 12295  df-z 12309  df-uz 12572  df-rp 12720  df-fz 13229  df-fzo 13372  df-hash 14034  df-word 14207  df-lsw 14255  df-concat 14263  df-s1 14290  df-substr 14343  df-pfx 14373  df-wwlks 28182  df-wwlksn 28183
This theorem is referenced by:  wwlksnextbij0  28253
  Copyright terms: Public domain W3C validator