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Theorem wwlksonvtx 29885
Description: If a word 𝑊 represents a walk of length 2 on two classes 𝐴 and 𝐶, these classes are vertices. (Contributed by AV, 14-Mar-2022.)
Hypothesis
Ref Expression
wwlksonvtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
wwlksonvtx (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → (𝐴𝑉𝐶𝑉))

Proof of Theorem wwlksonvtx
Dummy variables 𝑎 𝑏 𝑔 𝑛 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6920 . . . . 5 (Vtx‘𝑔) ∈ V
21, 1pm3.2i 470 . . . 4 ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
32rgen2w 3064 . . 3 𝑛 ∈ ℕ0𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
4 df-wwlksnon 29862 . . . 4 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
5 fveq2 6907 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
65, 5jca 511 . . . . 5 (𝑔 = 𝐺 → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
76adantl 481 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
84, 7el2mpocl 8110 . . 3 (∀𝑛 ∈ ℕ0𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))))
93, 8ax-mp 5 . 2 (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
10 wwlksonvtx.v . . . . 5 𝑉 = (Vtx‘𝐺)
1110eleq2i 2831 . . . 4 (𝐴𝑉𝐴 ∈ (Vtx‘𝐺))
1210eleq2i 2831 . . . 4 (𝐶𝑉𝐶 ∈ (Vtx‘𝐺))
1311, 12anbi12i 628 . . 3 ((𝐴𝑉𝐶𝑉) ↔ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
1413biimpri 228 . 2 ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴𝑉𝐶𝑉))
159, 14simpl2im 503 1 (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → (𝐴𝑉𝐶𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  cfv 6563  (class class class)co 7431  0cc0 11153  0cn0 12524  Vtxcvtx 29028   WWalksN cwwlksn 29856   WWalksNOn cwwlksnon 29857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-wwlksnon 29862
This theorem is referenced by:  iswspthsnon  29886  wwlks2onv  29983  elwwlks2ons3im  29984
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