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Theorem wspthnp 29539
Description: Properties of a set being a simple path of a fixed length as word. (Contributed by AV, 18-May-2021.)
Assertion
Ref Expression
wspthnp (π‘Š ∈ (𝑁 WSPathsN 𝐺) β†’ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š))
Distinct variable groups:   𝑓,𝐺   𝑓,π‘Š
Allowed substitution hint:   𝑁(𝑓)
Proof of Theorem wspthnp
Dummy variables 𝑔 𝑛 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wspthsn 29522 . . 3 WSPathsN = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ βˆƒπ‘“ 𝑓(SPathsβ€˜π‘”)𝑀})
21elmpocl 7652 . 2 (π‘Š ∈ (𝑁 WSPathsN 𝐺) β†’ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V))
3 simpl 482 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ π‘Š ∈ (𝑁 WSPathsN 𝐺)) β†’ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V))
4 iswspthn 29538 . . . . 5 (π‘Š ∈ (𝑁 WSPathsN 𝐺) ↔ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š))
54a1i 11 . . . 4 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (π‘Š ∈ (𝑁 WSPathsN 𝐺) ↔ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š)))
65biimpa 476 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ π‘Š ∈ (𝑁 WSPathsN 𝐺)) β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š))
7 3anass 1094 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š) ↔ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š)))
83, 6, 7sylanbrc 582 . 2 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ π‘Š ∈ (𝑁 WSPathsN 𝐺)) β†’ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š))
92, 8mpancom 685 1 (π‘Š ∈ (𝑁 WSPathsN 𝐺) β†’ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086  βˆƒwex 1780   ∈ wcel 2105  {crab 3431  Vcvv 3473   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  β„•0cn0 12479  SPathscspths 29405   WWalksN cwwlksn 29515   WSPathsN cwwspthsn 29517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-wwlksn 29520  df-wspthsn 29522
This theorem is referenced by:  wspthsswwlkn  29607
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