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Theorem wwlksonvtx 28969
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 6888 . . . . 5 (Vtx‘𝑔) ∈ V
21, 1pm3.2i 471 . . . 4 ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
32rgen2w 3065 . . 3 ∀𝑛 ∈ ℕ0 ∀𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
4 df-wwlksnon 28946 . . . 4 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↩ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↩ {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑛) = 𝑏)}))
5 fveq2 6875 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
65, 5jca 512 . . . . 5 (𝑔 = 𝐺 → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
76adantl 482 . . . 4 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
84, 7el2mpocl 8051 . . 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 2824 . . . 4 (𝐎 ∈ 𝑉 ↔ 𝐎 ∈ (Vtx‘𝐺))
1210eleq2i 2824 . . . 4 (𝐶 ∈ 𝑉 ↔ 𝐶 ∈ (Vtx‘𝐺))
1311, 12anbi12i 627 . . 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 504 1 (𝑊 ∈ (𝐎(𝑁 WWalksNOn 𝐺)𝐶) → (𝐎 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  âˆ€wral 3060  {crab 3429  Vcvv 3470  â€˜cfv 6529  (class class class)co 7390  0cc0 11089  â„•0cn0 12451  Vtxcvtx 28116   WWalksN cwwlksn 28940   WWalksNOn cwwlksnon 28941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7705
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-f1 6534  df-fo 6535  df-f1o 6536  df-fv 6537  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7954  df-2nd 7955  df-wwlksnon 28946
This theorem is referenced by:  iswspthsnon  28970  wwlks2onv  29067  elwwlks2ons3im  29068
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