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Theorem wwlksonvtx 27796
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 6690 . . . . 5 (Vtx‘𝑔) ∈ V
21, 1pm3.2i 474 . . . 4 ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
32rgen2w 3067 . . 3 𝑛 ∈ ℕ0𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
4 df-wwlksnon 27773 . . . 4 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
5 fveq2 6677 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
65, 5jca 515 . . . . 5 (𝑔 = 𝐺 → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
76adantl 485 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
84, 7el2mpocl 7810 . . 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 2825 . . . 4 (𝐴𝑉𝐴 ∈ (Vtx‘𝐺))
1210eleq2i 2825 . . . 4 (𝐶𝑉𝐶 ∈ (Vtx‘𝐺))
1311, 12anbi12i 630 . . 3 ((𝐴𝑉𝐶𝑉) ↔ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
1413biimpri 231 . 2 ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴𝑉𝐶𝑉))
159, 14simpl2im 507 1 (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → (𝐴𝑉𝐶𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  wral 3054  {crab 3058  Vcvv 3399  cfv 6340  (class class class)co 7173  0cc0 10618  0cn0 11979  Vtxcvtx 26944   WWalksN cwwlksn 27767   WWalksNOn cwwlksnon 27768
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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-1st 7717  df-2nd 7718  df-wwlksnon 27773
This theorem is referenced by:  iswspthsnon  27797  wwlks2onv  27894  elwwlks2ons3im  27895
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