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Theorem wspthsswwlkn 29703
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) βŠ† (𝑁 WWalksN 𝐺)

Proof of Theorem wspthsswwlkn
Dummy variables 𝑓 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wspthnp 29635 . . 3 (𝑀 ∈ (𝑁 WSPathsN 𝐺) β†’ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ 𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)𝑀))
21simp2d 1141 . 2 (𝑀 ∈ (𝑁 WSPathsN 𝐺) β†’ 𝑀 ∈ (𝑁 WWalksN 𝐺))
32ssriv 3982 1 (𝑁 WSPathsN 𝐺) βŠ† (𝑁 WWalksN 𝐺)
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 395  βˆƒwex 1774   ∈ wcel 2099  Vcvv 3469   βŠ† wss 3944   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  β„•0cn0 12488  SPathscspths 29501   WWalksN cwwlksn 29611   WSPathsN cwwspthsn 29613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-wwlksn 29616  df-wspthsn 29618
This theorem is referenced by:  wspthnfi  29704  fusgreg2wsp  30120
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