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Theorem wspthsswwlkn 29938
Description: The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length. (Contributed by AV, 18-May-2021.)
Assertion
Ref Expression
wspthsswwlkn (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺)

Proof of Theorem wspthsswwlkn
Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wspthnp 29870 . . 3 (𝑤 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ 𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑤))
21simp2d 1144 . 2 (𝑤 ∈ (𝑁 WSPathsN 𝐺) → 𝑤 ∈ (𝑁 WWalksN 𝐺))
32ssriv 3987 1 (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1779  wcel 2108  Vcvv 3480  wss 3951   class class class wbr 5143  cfv 6561  (class class class)co 7431  0cn0 12526  SPathscspths 29731   WWalksN cwwlksn 29846   WSPathsN cwwspthsn 29848
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-wwlksn 29851  df-wspthsn 29853
This theorem is referenced by:  wspthnfi  29939  fusgreg2wsp  30355
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