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

Theorem wwlksnextsurj 29421
Description: Lemma for wwlksnextbij 29423. (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 = (Vtxβ€˜πΊ)
wwlksnextbij0.e 𝐸 = (Edgβ€˜πΊ)
wwlksnextbij0.d 𝐷 = {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}
wwlksnextbij0.r 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸}
wwlksnextbij0.f 𝐹 = (𝑑 ∈ 𝐷 ↦ (lastSβ€˜π‘‘))
Assertion
Ref Expression
wwlksnextsurj (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ 𝐹:𝐷–onto→𝑅)
Distinct variable groups:   𝑀,𝐺   𝑀,𝑁   𝑀,π‘Š   𝑑,𝐷   𝑛,𝐸,𝑀   𝑑,𝑁,𝑀   𝑑,𝑅   𝑛,𝑉,𝑀   𝑛,π‘Š   𝑑,𝑛,𝑁,𝑀
Allowed substitution hints:   𝐷(𝑀,𝑛)   𝑅(𝑀,𝑛)   𝐸(𝑑)   𝐹(𝑀,𝑑,𝑛)   𝐺(𝑑,𝑛)   𝑉(𝑑)   π‘Š(𝑑)

Proof of Theorem wwlksnextsurj
Dummy variables 𝑖 𝑑 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlksnextbij0.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
21wwlknbp 29363 . . 3 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉))
3 simp2 1135 . . 3 ((𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉) β†’ 𝑁 ∈ β„•0)
4 wwlksnextbij0.e . . . 4 𝐸 = (Edgβ€˜πΊ)
5 wwlksnextbij0.d . . . 4 𝐷 = {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = (𝑁 + 2) ∧ (𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}
6 wwlksnextbij0.r . . . 4 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸}
7 wwlksnextbij0.f . . . 4 𝐹 = (𝑑 ∈ 𝐷 ↦ (lastSβ€˜π‘‘))
81, 4, 5, 6, 7wwlksnextfun 29419 . . 3 (𝑁 ∈ β„•0 β†’ 𝐹:π·βŸΆπ‘…)
92, 3, 83syl 18 . 2 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ 𝐹:π·βŸΆπ‘…)
10 preq2 4737 . . . . . 6 (𝑛 = π‘Ÿ β†’ {(lastSβ€˜π‘Š), 𝑛} = {(lastSβ€˜π‘Š), π‘Ÿ})
1110eleq1d 2816 . . . . 5 (𝑛 = π‘Ÿ β†’ ({(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸))
1211, 6elrab2 3685 . . . 4 (π‘Ÿ ∈ 𝑅 ↔ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸))
131, 4wwlksnext 29414 . . . . . . . . . . 11 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸) β†’ (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺))
14133expb 1118 . . . . . . . . . 10 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺))
15 s1cl 14556 . . . . . . . . . . . . . . . . . 18 (π‘Ÿ ∈ 𝑉 β†’ βŸ¨β€œπ‘Ÿβ€βŸ© ∈ Word 𝑉)
16 pfxccat1 14656 . . . . . . . . . . . . . . . . . 18 ((π‘Š ∈ Word 𝑉 ∧ βŸ¨β€œπ‘Ÿβ€βŸ© ∈ Word 𝑉) β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (β™―β€˜π‘Š)) = π‘Š)
1715, 16sylan2 591 . . . . . . . . . . . . . . . . 17 ((π‘Š ∈ Word 𝑉 ∧ π‘Ÿ ∈ 𝑉) β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (β™―β€˜π‘Š)) = π‘Š)
1817ex 411 . . . . . . . . . . . . . . . 16 (π‘Š ∈ Word 𝑉 β†’ (π‘Ÿ ∈ 𝑉 β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (β™―β€˜π‘Š)) = π‘Š))
1918adantr 479 . . . . . . . . . . . . . . 15 ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ (π‘Ÿ ∈ 𝑉 β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (β™―β€˜π‘Š)) = π‘Š))
20 oveq2 7419 . . . . . . . . . . . . . . . . . 18 ((𝑁 + 1) = (β™―β€˜π‘Š) β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (β™―β€˜π‘Š)))
2120eqcoms 2738 . . . . . . . . . . . . . . . . 17 ((β™―β€˜π‘Š) = (𝑁 + 1) β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (β™―β€˜π‘Š)))
2221eqeq1d 2732 . . . . . . . . . . . . . . . 16 ((β™―β€˜π‘Š) = (𝑁 + 1) β†’ (((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š ↔ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (β™―β€˜π‘Š)) = π‘Š))
2322adantl 480 . . . . . . . . . . . . . . 15 ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ (((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š ↔ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (β™―β€˜π‘Š)) = π‘Š))
2419, 23sylibrd 258 . . . . . . . . . . . . . 14 ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ (π‘Ÿ ∈ 𝑉 β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š))
25243adant3 1130 . . . . . . . . . . . . 13 ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸) β†’ (π‘Ÿ ∈ 𝑉 β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š))
261, 4wwlknp 29364 . . . . . . . . . . . . 13 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸))
2725, 26syl11 33 . . . . . . . . . . . 12 (π‘Ÿ ∈ 𝑉 β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š))
2827adantr 479 . . . . . . . . . . 11 ((π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸) β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š))
2928impcom 406 . . . . . . . . . 10 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š)
30 lswccats1 14588 . . . . . . . . . . . . . . . . . . 19 ((π‘Š ∈ Word 𝑉 ∧ π‘Ÿ ∈ 𝑉) β†’ (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)) = π‘Ÿ)
3130eqcomd 2736 . . . . . . . . . . . . . . . . . 18 ((π‘Š ∈ Word 𝑉 ∧ π‘Ÿ ∈ 𝑉) β†’ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)))
3231ex 411 . . . . . . . . . . . . . . . . 17 (π‘Š ∈ Word 𝑉 β†’ (π‘Ÿ ∈ 𝑉 β†’ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))))
33323ad2ant3 1133 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉) β†’ (π‘Ÿ ∈ 𝑉 β†’ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))))
342, 33syl 17 . . . . . . . . . . . . . . 15 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (π‘Ÿ ∈ 𝑉 β†’ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))))
3534imp 405 . . . . . . . . . . . . . 14 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ π‘Ÿ ∈ 𝑉) β†’ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)))
3635preq2d 4743 . . . . . . . . . . . . 13 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ π‘Ÿ ∈ 𝑉) β†’ {(lastSβ€˜π‘Š), π‘Ÿ} = {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))})
3736eleq1d 2816 . . . . . . . . . . . 12 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ π‘Ÿ ∈ 𝑉) β†’ ({(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸))
3837biimpd 228 . . . . . . . . . . 11 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ π‘Ÿ ∈ 𝑉) β†’ ({(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸 β†’ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸))
3938impr 453 . . . . . . . . . 10 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸)
4014, 29, 39jca32 514 . . . . . . . . 9 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸)))
4133, 2syl11 33 . . . . . . . . . . 11 (π‘Ÿ ∈ 𝑉 β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))))
4241adantr 479 . . . . . . . . . 10 ((π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸) β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))))
4342impcom 406 . . . . . . . . 9 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)))
44 ovexd 7446 . . . . . . . . . 10 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) ∈ V)
45 eleq1 2819 . . . . . . . . . . . . . . 15 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ (𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺)))
46 oveq1 7418 . . . . . . . . . . . . . . . . 17 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ (𝑑 prefix (𝑁 + 1)) = ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)))
4746eqeq1d 2732 . . . . . . . . . . . . . . . 16 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ ((𝑑 prefix (𝑁 + 1)) = π‘Š ↔ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š))
48 fveq2 6890 . . . . . . . . . . . . . . . . . 18 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ (lastSβ€˜π‘‘) = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)))
4948preq2d 4743 . . . . . . . . . . . . . . . . 17 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} = {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))})
5049eleq1d 2816 . . . . . . . . . . . . . . . 16 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ ({(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸))
5147, 50anbi12d 629 . . . . . . . . . . . . . . 15 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ (((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸) ↔ (((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸)))
5245, 51anbi12d 629 . . . . . . . . . . . . . 14 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ↔ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸))))
5348eqeq2d 2741 . . . . . . . . . . . . . 14 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ (π‘Ÿ = (lastSβ€˜π‘‘) ↔ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))))
5452, 53anbi12d 629 . . . . . . . . . . . . 13 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ (((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜π‘‘)) ↔ (((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)))))
5554bicomd 222 . . . . . . . . . . . 12 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ ((((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜π‘‘))))
5655adantl 480 . . . . . . . . . . 11 (((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) ∧ 𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)) β†’ ((((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜π‘‘))))
5756biimpd 228 . . . . . . . . . 10 (((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) ∧ 𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)) β†’ ((((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))) β†’ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜π‘‘))))
5844, 57spcimedv 3584 . . . . . . . . 9 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ ((((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))) β†’ βˆƒπ‘‘((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜π‘‘))))
5940, 43, 58mp2and 695 . . . . . . . 8 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ βˆƒπ‘‘((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜π‘‘)))
60 oveq1 7418 . . . . . . . . . . . . 13 (𝑀 = 𝑑 β†’ (𝑀 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
6160eqeq1d 2732 . . . . . . . . . . . 12 (𝑀 = 𝑑 β†’ ((𝑀 prefix (𝑁 + 1)) = π‘Š ↔ (𝑑 prefix (𝑁 + 1)) = π‘Š))
62 fveq2 6890 . . . . . . . . . . . . . 14 (𝑀 = 𝑑 β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘‘))
6362preq2d 4743 . . . . . . . . . . . . 13 (𝑀 = 𝑑 β†’ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} = {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)})
6463eleq1d 2816 . . . . . . . . . . . 12 (𝑀 = 𝑑 β†’ ({(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸))
6561, 64anbi12d 629 . . . . . . . . . . 11 (𝑀 = 𝑑 β†’ (((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸) ↔ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)))
6665elrab 3682 . . . . . . . . . 10 (𝑑 ∈ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} ↔ (𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)))
6766anbi1i 622 . . . . . . . . 9 ((𝑑 ∈ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} ∧ π‘Ÿ = (lastSβ€˜π‘‘)) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜π‘‘)))
6867exbii 1848 . . . . . . . 8 (βˆƒπ‘‘(𝑑 ∈ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} ∧ π‘Ÿ = (lastSβ€˜π‘‘)) ↔ βˆƒπ‘‘((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸)) ∧ π‘Ÿ = (lastSβ€˜π‘‘)))
6959, 68sylibr 233 . . . . . . 7 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ βˆƒπ‘‘(𝑑 ∈ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} ∧ π‘Ÿ = (lastSβ€˜π‘‘)))
70 df-rex 3069 . . . . . . 7 (βˆƒπ‘‘ ∈ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}π‘Ÿ = (lastSβ€˜π‘‘) ↔ βˆƒπ‘‘(𝑑 ∈ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)} ∧ π‘Ÿ = (lastSβ€˜π‘‘)))
7169, 70sylibr 233 . . . . . 6 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ βˆƒπ‘‘ ∈ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}π‘Ÿ = (lastSβ€˜π‘‘))
721, 4, 5wwlksnextwrd 29418 . . . . . . . 8 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ 𝐷 = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)})
7372adantr 479 . . . . . . 7 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ 𝐷 = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)})
7473rexeqdv 3324 . . . . . 6 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ (βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (lastSβ€˜π‘‘) ↔ βˆƒπ‘‘ ∈ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = π‘Š ∧ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸)}π‘Ÿ = (lastSβ€˜π‘‘)))
7571, 74mpbird 256 . . . . 5 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (lastSβ€˜π‘‘))
76 fveq2 6890 . . . . . . . 8 (𝑑 = 𝑑 β†’ (lastSβ€˜π‘‘) = (lastSβ€˜π‘‘))
77 fvex 6903 . . . . . . . 8 (lastSβ€˜π‘‘) ∈ V
7876, 7, 77fvmpt 6997 . . . . . . 7 (𝑑 ∈ 𝐷 β†’ (πΉβ€˜π‘‘) = (lastSβ€˜π‘‘))
7978eqeq2d 2741 . . . . . 6 (𝑑 ∈ 𝐷 β†’ (π‘Ÿ = (πΉβ€˜π‘‘) ↔ π‘Ÿ = (lastSβ€˜π‘‘)))
8079rexbiia 3090 . . . . 5 (βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (πΉβ€˜π‘‘) ↔ βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (lastSβ€˜π‘‘))
8175, 80sylibr 233 . . . 4 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (πΉβ€˜π‘‘))
8212, 81sylan2b 592 . . 3 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ π‘Ÿ ∈ 𝑅) β†’ βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (πΉβ€˜π‘‘))
8382ralrimiva 3144 . 2 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ βˆ€π‘Ÿ ∈ 𝑅 βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (πΉβ€˜π‘‘))
84 dffo3 7102 . 2 (𝐹:𝐷–onto→𝑅 ↔ (𝐹:π·βŸΆπ‘… ∧ βˆ€π‘Ÿ ∈ 𝑅 βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (πΉβ€˜π‘‘)))
859, 83, 84sylanbrc 581 1 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ 𝐹:𝐷–onto→𝑅)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430  Vcvv 3472  {cpr 4629   ↦ cmpt 5230  βŸΆwf 6538  β€“ontoβ†’wfo 6540  β€˜cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115  2c2 12271  β„•0cn0 12476  ..^cfzo 13631  β™―chash 14294  Word cword 14468  lastSclsw 14516   ++ cconcat 14524  βŸ¨β€œcs1 14549   prefix cpfx 14624  Vtxcvtx 28523  Edgcedg 28574   WWalksN cwwlksn 29347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-rp 12979  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-lsw 14517  df-concat 14525  df-s1 14550  df-substr 14595  df-pfx 14625  df-wwlks 29351  df-wwlksn 29352
This theorem is referenced by:  wwlksnextbij0  29422
  Copyright terms: Public domain W3C validator