Theorem wwlksonvtx 28121

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) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))

Proof of Theorem wwlksonvtx

Dummy variables 𝑎 𝑏 𝑔 𝑛 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6769	. . . . 5 ⊢ (Vtx‘𝑔) ∈ V
2	1, 1	pm3.2i 470	. . . 4 ⊢ ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
3	2	rgen2w 3076	. . 3 ⊢ ∀𝑛 ∈ ℕ0 ∀𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
4		df-wwlksnon 28098	. . . 4 ⊢ WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}))
5		fveq2 6756	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
6	5, 5	jca 511	. . . . 5 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
7	6	adantl 481	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
8	4, 7	el2mpocl 7897	. . 3 ⊢ (∀𝑛 ∈ ℕ0 ∀𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))))
9	3, 8	ax-mp 5	. 2 ⊢ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
10		wwlksonvtx.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
11	10	eleq2i 2830	. . . 4 ⊢ (𝐴 ∈ 𝑉 ↔ 𝐴 ∈ (Vtx‘𝐺))
12	10	eleq2i 2830	. . . 4 ⊢ (𝐶 ∈ 𝑉 ↔ 𝐶 ∈ (Vtx‘𝐺))
13	11, 12	anbi12i 626	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ↔ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
14	13	biimpri 227	. 2 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
15	9, 14	simpl2im 503	1 ⊢ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422  ‘cfv 6418  (class class class)co 7255  0cc0 10802  ℕ₀cn0 12163  Vtxcvtx 27269   WWalksN cwwlksn 28092   WWalksNOn cwwlksnon 28093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-wwlksnon 28098

This theorem is referenced by:  iswspthsnon  28122  wwlks2onv  28219  elwwlks2ons3im  28220