Theorem wwlksonvtx 29708

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 6904	. . . . 5 ⊢ (Vtx‘𝑔) ∈ V
2	1, 1	pm3.2i 469	. . . 4 ⊢ ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
3	2	rgen2w 3056	. . 3 ⊢ ∀𝑛 ∈ ℕ0 ∀𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
4		df-wwlksnon 29685	. . . 4 ⊢ WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}))
5		fveq2 6891	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
6	5, 5	jca 510	. . . . 5 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
7	6	adantl 480	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
8	4, 7	el2mpocl 8087	. . 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 2817	. . . 4 ⊢ (𝐴 ∈ 𝑉 ↔ 𝐴 ∈ (Vtx‘𝐺))
12	10	eleq2i 2817	. . . 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 502	1 ⊢ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3051  {crab 3419  Vcvv 3463  ‘cfv 6542  (class class class)co 7415  0cc0 11136  ℕ₀cn0 12500  Vtxcvtx 28851   WWalksN cwwlksn 29679   WWalksNOn cwwlksnon 29680

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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-wwlksnon 29685

This theorem is referenced by:  iswspthsnon  29709  wwlks2onv  29806  elwwlks2ons3im  29807