Theorem wspthsswwlkn 29951

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.

Step	Hyp	Ref	Expression
1		wspthnp 29883	. . 3 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑤))
2	1	simp2d 1143	. 2 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → 𝑤 ∈ (𝑁 WWalksN 𝐺))
3	2	ssriv 4012	1 ⊢ (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺)

Syntax hints:   ∧ wa 395  ∃wex 1777   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  ℕ₀cn0 12553  SPathscspths 29749   WWalksN cwwlksn 29859   WSPathsN cwwspthsn 29861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-wwlksn 29864  df-wspthsn 29866

This theorem is referenced by:  wspthnfi  29952  fusgreg2wsp  30368