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Theorem extwwlkfab 30114
Description: The set (𝑋𝐢𝑁) of double loops of length 𝑁 on vertex 𝑋 can be constructed from the set 𝐹 of closed walks on 𝑋 with length smaller by 2 than the fixed length by appending a neighbor of the last vertex and afterwards the last vertex (which is the first vertex) itself ("walking forth and back" from the last vertex). 3 ≀ 𝑁 is required since for 𝑁 = 2: 𝐹 = (𝑋(ClWWalksNOnβ€˜πΊ)0) = βˆ… (see clwwlk0on0 29854 stating that a closed walk of length 0 is not represented as word), which would result in an empty set on the right hand side, but (𝑋𝐢𝑁) needs not be empty, see 2clwwlk2 30110. (Contributed by Alexander van der Vekens, 18-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Oct-2022.)
Hypotheses
Ref Expression
extwwlkfab.v 𝑉 = (Vtxβ€˜πΊ)
extwwlkfab.c 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
extwwlkfab.f 𝐹 = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))
Assertion
Ref Expression
extwwlkfab ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)})
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣,𝑀   𝑛,𝑋,𝑣,𝑀
Allowed substitution hints:   𝐢(𝑀,𝑣,𝑛)   𝐹(𝑀,𝑣,𝑛)

Proof of Theorem extwwlkfab
StepHypRef Expression
1 uzuzle23 12877 . . . 4 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ (β„€β‰₯β€˜2))
2 extwwlkfab.c . . . . 5 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
322clwwlk 30109 . . . 4 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋})
41, 3sylan2 592 . . 3 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋})
543adant1 1127 . 2 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋})
6 clwwlknon 29852 . . . 4 (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) = {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}
76rabeqi 3439 . . 3 {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋} = {𝑀 ∈ {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋}
8 rabrab 3449 . . . 4 {𝑀 ∈ {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋} = {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)}
9 simpll3 1211 . . . . . . . . . . 11 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ 𝑁 ∈ (β„€β‰₯β€˜3))
10 simplr 766 . . . . . . . . . . 11 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ 𝑀 ∈ (𝑁 ClWWalksN 𝐺))
11 simpr 484 . . . . . . . . . . . . 13 (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)
12 simpl 482 . . . . . . . . . . . . . 14 (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘€β€˜0) = 𝑋)
1312eqcomd 2732 . . . . . . . . . . . . 13 (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ 𝑋 = (π‘€β€˜0))
1411, 13eqtrd 2766 . . . . . . . . . . . 12 (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘€β€˜(𝑁 βˆ’ 2)) = (π‘€β€˜0))
1514adantl 481 . . . . . . . . . . 11 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (π‘€β€˜(𝑁 βˆ’ 2)) = (π‘€β€˜0))
16 clwwnrepclwwn 30106 . . . . . . . . . . 11 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = (π‘€β€˜0)) β†’ (𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺))
179, 10, 15, 16syl3anc 1368 . . . . . . . . . 10 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺))
1812adantl 481 . . . . . . . . . 10 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (π‘€β€˜0) = 𝑋)
1917, 18jca 511 . . . . . . . . 9 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋))
20 simp1 1133 . . . . . . . . . . . . 13 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝐺 ∈ USGraph)
2120anim1i 614 . . . . . . . . . . . 12 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (𝐺 ∈ USGraph ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)))
2221adantr 480 . . . . . . . . . . 11 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (𝐺 ∈ USGraph ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)))
23 clwwlknlbonbgr1 29801 . . . . . . . . . . 11 ((𝐺 ∈ USGraph ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx (π‘€β€˜0)))
2422, 23syl 17 . . . . . . . . . 10 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx (π‘€β€˜0)))
25 oveq2 7413 . . . . . . . . . . . . 13 (𝑋 = (π‘€β€˜0) β†’ (𝐺 NeighbVtx 𝑋) = (𝐺 NeighbVtx (π‘€β€˜0)))
2625eqcoms 2734 . . . . . . . . . . . 12 ((π‘€β€˜0) = 𝑋 β†’ (𝐺 NeighbVtx 𝑋) = (𝐺 NeighbVtx (π‘€β€˜0)))
2726adantr 480 . . . . . . . . . . 11 (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (𝐺 NeighbVtx 𝑋) = (𝐺 NeighbVtx (π‘€β€˜0)))
2827adantl 481 . . . . . . . . . 10 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (𝐺 NeighbVtx 𝑋) = (𝐺 NeighbVtx (π‘€β€˜0)))
2924, 28eleqtrrd 2830 . . . . . . . . 9 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋))
3011adantl 481 . . . . . . . . 9 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)
3119, 29, 303jca 1125 . . . . . . . 8 ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) ∧ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋))
3231ex 412 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)))
33 simpr 484 . . . . . . . . 9 (((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) β†’ (π‘€β€˜0) = 𝑋)
3433anim1i 614 . . . . . . . 8 ((((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋))
35343adant2 1128 . . . . . . 7 ((((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋))
3632, 35impbid1 224 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) ↔ (((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)))
37 2clwwlklem 30105 . . . . . . . . . . . 12 ((𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = (π‘€β€˜0))
38373ad2antr3 1187 . . . . . . . . . . 11 ((𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = (π‘€β€˜0))
3938ancoms 458 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = (π‘€β€˜0))
4039eqcomd 2732 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (π‘€β€˜0) = ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0))
4140eqeq1d 2728 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ ((π‘€β€˜0) = 𝑋 ↔ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = 𝑋))
4241anbi2d 628 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) ↔ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = 𝑋)))
43423anbi1d 1436 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ ((((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) ↔ (((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)))
44 extwwlkfab.f . . . . . . . . . . 11 𝐹 = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))
4544eleq2i 2819 . . . . . . . . . 10 ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ↔ (𝑀 prefix (𝑁 βˆ’ 2)) ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))
46 isclwwlknon 29853 . . . . . . . . . . 11 ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)) ↔ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = 𝑋))
4746a1i 11 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)) ↔ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = 𝑋)))
4845, 47bitrid 283 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ↔ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = 𝑋)))
49483anbi1d 1436 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) ↔ (((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)))
5049bicomd 222 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) ↔ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)))
5150adantr 480 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ ((((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) ↔ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)))
5236, 43, 513bitrd 305 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) ↔ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)))
5352rabbidva 3433 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)} = {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)})
548, 53eqtrid 2778 . . 3 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ {𝑀 ∈ {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋} = {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)})
557, 54eqtrid 2778 . 2 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋} = {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)})
565, 55eqtrd 2766 1 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3426  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  0cc0 11112  1c1 11113   βˆ’ cmin 11448  2c2 12271  3c3 12272  β„€β‰₯cuz 12826   prefix cpfx 14626  Vtxcvtx 28764  USGraphcusgr 28917   NeighbVtx cnbgr 29097   ClWWalksN cclwwlkn 29786  ClWWalksNOncclwwlknon 29849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  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 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-oadd 8471  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  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-3 12280  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-lsw 14519  df-substr 14597  df-pfx 14627  df-edg 28816  df-upgr 28850  df-umgr 28851  df-usgr 28919  df-nbgr 29098  df-wwlks 29593  df-wwlksn 29594  df-clwwlk 29744  df-clwwlkn 29787  df-clwwlknon 29850
This theorem is referenced by:  extwwlkfabel  30115
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