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Theorem wwlksnextsurj 29154
Description: Lemma for wwlksnextbij 29156. (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 29096 . . 3 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉))
3 simp2 1138 . . 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 29152 . . 3 (𝑁 ∈ β„•0 β†’ 𝐹:π·βŸΆπ‘…)
92, 3, 83syl 18 . 2 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ 𝐹:π·βŸΆπ‘…)
10 preq2 4739 . . . . . 6 (𝑛 = π‘Ÿ β†’ {(lastSβ€˜π‘Š), 𝑛} = {(lastSβ€˜π‘Š), π‘Ÿ})
1110eleq1d 2819 . . . . 5 (𝑛 = π‘Ÿ β†’ ({(lastSβ€˜π‘Š), 𝑛} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸))
1211, 6elrab2 3687 . . . 4 (π‘Ÿ ∈ 𝑅 ↔ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸))
131, 4wwlksnext 29147 . . . . . . . . . . 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 1121 . . . . . . . . . 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 14552 . . . . . . . . . . . . . . . . . 18 (π‘Ÿ ∈ 𝑉 β†’ βŸ¨β€œπ‘Ÿβ€βŸ© ∈ Word 𝑉)
16 pfxccat1 14652 . . . . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . . . . . 17 ((π‘Š ∈ Word 𝑉 ∧ π‘Ÿ ∈ 𝑉) β†’ ((π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) prefix (β™―β€˜π‘Š)) = π‘Š)
1817ex 414 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 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 7417 . . . . . . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . 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 259 . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . 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 29097 . . . . . . . . . . . . 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 482 . . . . . . . . . . 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 409 . . . . . . . . . 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 14584 . . . . . . . . . . . . . . . . . . 19 ((π‘Š ∈ Word 𝑉 ∧ π‘Ÿ ∈ 𝑉) β†’ (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)) = π‘Ÿ)
3130eqcomd 2739 . . . . . . . . . . . . . . . . . 18 ((π‘Š ∈ Word 𝑉 ∧ π‘Ÿ ∈ 𝑉) β†’ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)))
3231ex 414 . . . . . . . . . . . . . . . . 17 (π‘Š ∈ Word 𝑉 β†’ (π‘Ÿ ∈ 𝑉 β†’ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©))))
33323ad2ant3 1136 . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . 14 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ π‘Ÿ ∈ 𝑉) β†’ π‘Ÿ = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)))
3635preq2d 4745 . . . . . . . . . . . . 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 2819 . . . . . . . . . . . 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 456 . . . . . . . . . 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 517 . . . . . . . . 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 482 . . . . . . . . . 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 409 . . . . . . . . 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 7444 . . . . . . . . . 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 2822 . . . . . . . . . . . . . . 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 7416 . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . 18 (𝑑 = (π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©) β†’ (lastSβ€˜π‘‘) = (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘Ÿβ€βŸ©)))
4948preq2d 4745 . . . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . 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 2744 . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . 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 483 . . . . . . . . . . 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 3586 . . . . . . . . 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 698 . . . . . . . 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 7416 . . . . . . . . . . . . 13 (𝑀 = 𝑑 β†’ (𝑀 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
6160eqeq1d 2735 . . . . . . . . . . . 12 (𝑀 = 𝑑 β†’ ((𝑀 prefix (𝑁 + 1)) = π‘Š ↔ (𝑑 prefix (𝑁 + 1)) = π‘Š))
62 fveq2 6892 . . . . . . . . . . . . . 14 (𝑀 = 𝑑 β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘‘))
6362preq2d 4745 . . . . . . . . . . . . 13 (𝑀 = 𝑑 β†’ {(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} = {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)})
6463eleq1d 2819 . . . . . . . . . . . 12 (𝑀 = 𝑑 β†’ ({(lastSβ€˜π‘Š), (lastSβ€˜π‘€)} ∈ 𝐸 ↔ {(lastSβ€˜π‘Š), (lastSβ€˜π‘‘)} ∈ 𝐸))
6561, 64anbi12d 632 . . . . . . . . . . 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 3684 . . . . . . . . . 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 625 . . . . . . . . 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 1851 . . . . . . . 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 3072 . . . . . . 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 29151 . . . . . . . 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 482 . . . . . . 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 3327 . . . . . 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 257 . . . . 5 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (π‘Ÿ ∈ 𝑉 ∧ {(lastSβ€˜π‘Š), π‘Ÿ} ∈ 𝐸)) β†’ βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (lastSβ€˜π‘‘))
76 fveq2 6892 . . . . . . . 8 (𝑑 = 𝑑 β†’ (lastSβ€˜π‘‘) = (lastSβ€˜π‘‘))
77 fvex 6905 . . . . . . . 8 (lastSβ€˜π‘‘) ∈ V
7876, 7, 77fvmpt 6999 . . . . . . 7 (𝑑 ∈ 𝐷 β†’ (πΉβ€˜π‘‘) = (lastSβ€˜π‘‘))
7978eqeq2d 2744 . . . . . 6 (𝑑 ∈ 𝐷 β†’ (π‘Ÿ = (πΉβ€˜π‘‘) ↔ π‘Ÿ = (lastSβ€˜π‘‘)))
8079rexbiia 3093 . . . . 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 595 . . 3 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ π‘Ÿ ∈ 𝑅) β†’ βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (πΉβ€˜π‘‘))
8382ralrimiva 3147 . 2 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ βˆ€π‘Ÿ ∈ 𝑅 βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (πΉβ€˜π‘‘))
84 dffo3 7104 . 2 (𝐹:𝐷–onto→𝑅 ↔ (𝐹:π·βŸΆπ‘… ∧ βˆ€π‘Ÿ ∈ 𝑅 βˆƒπ‘‘ ∈ 𝐷 π‘Ÿ = (πΉβ€˜π‘‘)))
859, 83, 84sylanbrc 584 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 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475  {cpr 4631   ↦ cmpt 5232  βŸΆwf 6540  β€“ontoβ†’wfo 6542  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113  2c2 12267  β„•0cn0 12472  ..^cfzo 13627  β™―chash 14290  Word cword 14464  lastSclsw 14512   ++ cconcat 14520  βŸ¨β€œcs1 14545   prefix cpfx 14620  Vtxcvtx 28256  Edgcedg 28307   WWalksN cwwlksn 29080
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-lsw 14513  df-concat 14521  df-s1 14546  df-substr 14591  df-pfx 14621  df-wwlks 29084  df-wwlksn 29085
This theorem is referenced by:  wwlksnextbij0  29155
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