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Theorem extwwlkfab 29338
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 29078 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 29334. (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 12821 . . . 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 29333 . . . 4 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋})
41, 3sylan2 594 . . 3 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋})
543adant1 1131 . 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 29076 . . . 4 (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) = {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}
76rabeqi 3423 . . 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 3433 . . . 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 1215 . . . . . . . . . . 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 768 . . . . . . . . . . 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 486 . . . . . . . . . . . . 13 (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)
12 simpl 484 . . . . . . . . . . . . . 14 (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘€β€˜0) = 𝑋)
1312eqcomd 2743 . . . . . . . . . . . . 13 (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ 𝑋 = (π‘€β€˜0))
1411, 13eqtrd 2777 . . . . . . . . . . . 12 (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘€β€˜(𝑁 βˆ’ 2)) = (π‘€β€˜0))
1514adantl 483 . . . . . . . . . . 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 29330 . . . . . . . . . . 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 1372 . . . . . . . . . 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 483 . . . . . . . . . 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 513 . . . . . . . . 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 1137 . . . . . . . . . . . . 13 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝐺 ∈ USGraph)
2120anim1i 616 . . . . . . . . . . . 12 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (𝐺 ∈ USGraph ∧ 𝑀 ∈ (𝑁 ClWWalksN 𝐺)))
2221adantr 482 . . . . . . . . . . 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 29025 . . . . . . . . . . 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 7370 . . . . . . . . . . . . 13 (𝑋 = (π‘€β€˜0) β†’ (𝐺 NeighbVtx 𝑋) = (𝐺 NeighbVtx (π‘€β€˜0)))
2625eqcoms 2745 . . . . . . . . . . . 12 ((π‘€β€˜0) = 𝑋 β†’ (𝐺 NeighbVtx 𝑋) = (𝐺 NeighbVtx (π‘€β€˜0)))
2726adantr 482 . . . . . . . . . . 11 (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (𝐺 NeighbVtx 𝑋) = (𝐺 NeighbVtx (π‘€β€˜0)))
2827adantl 483 . . . . . . . . . 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 2841 . . . . . . . . 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 483 . . . . . . . . 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 1129 . . . . . . . 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 414 . . . . . . 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 486 . . . . . . . . 9 (((𝑀 prefix (𝑁 βˆ’ 2)) ∈ ((𝑁 βˆ’ 2) ClWWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑋) β†’ (π‘€β€˜0) = 𝑋)
3433anim1i 616 . . . . . . . 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 1132 . . . . . . 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 29329 . . . . . . . . . . . 12 ((𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((𝑀 prefix (𝑁 βˆ’ 2))β€˜0) = (π‘€β€˜0))
38373ad2antr3 1191 . . . . . . . . . . 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 460 . . . . . . . . . 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 2743 . . . . . . . . 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 2739 . . . . . . . 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 630 . . . . . . 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 1441 . . . . . 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 2830 . . . . . . . . . 10 ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ↔ (𝑀 prefix (𝑁 βˆ’ 2)) ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))
46 isclwwlknon 29077 . . . . . . . . . . 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 1441 . . . . . . . 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 482 . . . . . 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 3417 . . . 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 2789 . . 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 2789 . 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 2777 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3410  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  0cc0 11058  1c1 11059   βˆ’ cmin 11392  2c2 12215  3c3 12216  β„€β‰₯cuz 12770   prefix cpfx 14565  Vtxcvtx 27989  USGraphcusgr 28142   NeighbVtx cnbgr 28322   ClWWalksN cclwwlkn 29010  ClWWalksNOncclwwlknon 29073
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-lsw 14458  df-substr 14536  df-pfx 14566  df-edg 28041  df-upgr 28075  df-umgr 28076  df-usgr 28144  df-nbgr 28323  df-wwlks 28817  df-wwlksn 28818  df-clwwlk 28968  df-clwwlkn 29011  df-clwwlknon 29074
This theorem is referenced by:  extwwlkfabel  29339
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