Theorem wspthsswwlkn 30004

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 29936	. . 3 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑤))
2	1	simp2d 1149	. 2 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → 𝑤 ∈ (𝑁 WWalksN 𝐺))
3	2	ssriv 3919	1 ⊢ (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺)

Syntax hints:   ∧ wa 396  ∃wex 1786   ∈ wcel 2119  Vcvv 3431   ⊆ wss 3883   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  ℕ₀cn0 12428  SPathscspths 29797   WWalksN cwwlksn 29912   WSPathsN cwwspthsn 29914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-wwlksn 29917  df-wspthsn 29919

This theorem is referenced by:  wspthnfi  30005  fusgreg2wsp  30424