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Theorem wspthsswwlkn 29768
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 29700 . . 3 (𝑀 ∈ (𝑁 WSPathsN 𝐺) β†’ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ 𝑀 ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)𝑀))
21simp2d 1140 . 2 (𝑀 ∈ (𝑁 WSPathsN 𝐺) β†’ 𝑀 ∈ (𝑁 WWalksN 𝐺))
32ssriv 3977 1 (𝑁 WSPathsN 𝐺) βŠ† (𝑁 WWalksN 𝐺)
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 394  βˆƒwex 1773   ∈ wcel 2098  Vcvv 3463   βŠ† wss 3941   class class class wbr 5144  β€˜cfv 6543  (class class class)co 7413  β„•0cn0 12497  SPathscspths 29566   WWalksN cwwlksn 29676   WSPathsN cwwspthsn 29678
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 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-wwlksn 29681  df-wspthsn 29683
This theorem is referenced by:  wspthnfi  29769  fusgreg2wsp  30185
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