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Theorem wwlksonvtx 29618
Description: If a word 𝑊 represents a walk of length 2 on two classes 𝐎 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 ∈ (𝐎(𝑁 WWalksNOn 𝐺)𝐶) → (𝐎 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))

Proof of Theorem wwlksonvtx
Dummy variables 𝑎 𝑏 𝑔 𝑛 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6898 . . . . 5 (Vtx‘𝑔) ∈ V
21, 1pm3.2i 470 . . . 4 ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
32rgen2w 3060 . . 3 ∀𝑛 ∈ ℕ0 ∀𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
4 df-wwlksnon 29595 . . . 4 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↩ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↩ {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑛) = 𝑏)}))
5 fveq2 6885 . . . . . 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 = 𝑁 ∧ 𝑔 = 𝐺) → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
84, 7el2mpocl 8072 . . 3 (∀𝑛 ∈ ℕ0 ∀𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V) → (𝑊 ∈ (𝐎(𝑁 WWalksNOn 𝐺)𝐶) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))))
93, 8ax-mp 5 . 2 (𝑊 ∈ (𝐎(𝑁 WWalksNOn 𝐺)𝐶) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
10 wwlksonvtx.v . . . . 5 𝑉 = (Vtx‘𝐺)
1110eleq2i 2819 . . . 4 (𝐎 ∈ 𝑉 ↔ 𝐎 ∈ (Vtx‘𝐺))
1210eleq2i 2819 . . . 4 (𝐶 ∈ 𝑉 ↔ 𝐶 ∈ (Vtx‘𝐺))
1311, 12anbi12i 626 . . 3 ((𝐎 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ↔ (𝐎 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
1413biimpri 227 . 2 ((𝐎 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐎 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
159, 14simpl2im 503 1 (𝑊 ∈ (𝐎(𝑁 WWalksNOn 𝐺)𝐶) → (𝐎 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  âˆ€wral 3055  {crab 3426  Vcvv 3468  â€˜cfv 6537  (class class class)co 7405  0cc0 11112  â„•0cn0 12476  Vtxcvtx 28764   WWalksN cwwlksn 29589   WWalksNOn cwwlksnon 29590
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-wwlksnon 29595
This theorem is referenced by:  iswspthsnon  29619  wwlks2onv  29716  elwwlks2ons3im  29717
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