Theorem wspthnonp 29879

Description: Properties of a set being a simple path of a fixed length between two vertices as word. (Contributed by AV, 14-May-2021.) (Proof shortened by AV, 15-Mar-2022.)

Hypothesis

Ref	Expression
wspthnonp.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
wspthnonp	⊢ (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑓,𝐺   𝑓,𝑁   𝑓,𝑉   𝑓,𝑊

Proof of Theorem wspthnonp

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

Step	Hyp	Ref	Expression
1		fvex 6919	. . . . 5 ⊢ (Vtx‘𝑔) ∈ V
2	1, 1	pm3.2i 470	. . . 4 ⊢ ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
3	2	rgen2w 3066	. . 3 ⊢ ∀𝑛 ∈ ℕ0 ∀𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
4		df-wspthsnon 29854	. . . 4 ⊢ WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
5		fveq2 6906	. . . . . 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 8111	. . 3 ⊢ (∀𝑛 ∈ ℕ0 ∀𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V) → (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))))
9	3, 8	ax-mp 5	. 2 ⊢ (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
10		simprl 771	. . 3 ⊢ ((𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∧ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))) → (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V))
11		wspthnonp.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
12	11	eleq2i 2833	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 ↔ 𝐴 ∈ (Vtx‘𝐺))
13	11	eleq2i 2833	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 ↔ 𝐵 ∈ (Vtx‘𝐺))
14	12, 13	anbi12i 628	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ↔ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
15	14	biimpri 228	. . . . 5 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
16	15	adantl 481	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
17	16	adantl 481	. . 3 ⊢ ((𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∧ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
18		wspthnon 29878	. . . . 5 ⊢ (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
19	18	biimpi 216	. . . 4 ⊢ (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
20	19	adantr 480	. . 3 ⊢ ((𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∧ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
21	10, 17, 20	3jca 1129	. 2 ⊢ ((𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∧ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))
22	9, 21	mpdan 687	1 ⊢ (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  {crab 3436  Vcvv 3480   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  ℕ₀cn0 12526  Vtxcvtx 29013  SPathsOncspthson 29733   WWalksNOn cwwlksnon 29847   WSPathsNOn cwwspthsnon 29849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-wwlksnon 29852  df-wspthsnon 29854

This theorem is referenced by:  wspthneq1eq2  29880  wspthsnonn0vne  29937  wspthsswwlknon  29941