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Theorem wspthnonp 26813
 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
wwlknon.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
wspthnonp (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ (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.
StepHypRef Expression
1 fvex 6239 . . . . 5 (Vtx‘𝑔) ∈ V
21, 1pm3.2i 470 . . . 4 ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
32rgen2w 2954 . . 3 𝑛 ∈ ℕ0𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V)
4 df-wspthsnon 26782 . . . 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 6229 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
65, 5jca 553 . . . . 5 (𝑔 = 𝐺 → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
76adantl 481 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺)))
84, 7el2mpt2cl 7296 . . 3 (∀𝑛 ∈ ℕ0𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V) → (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))))
93, 8ax-mp 5 . 2 (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
10 simprl 809 . . 3 ((𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∧ ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))) → (𝑁 ∈ ℕ0𝐺 ∈ V))
11 wwlknon.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
1211eleq2i 2722 . . . . . . 7 (𝐴𝑉𝐴 ∈ (Vtx‘𝐺))
1311eleq2i 2722 . . . . . . 7 (𝐵𝑉𝐵 ∈ (Vtx‘𝐺))
1412, 13anbi12i 733 . . . . . 6 ((𝐴𝑉𝐵𝑉) ↔ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
1514biimpri 218 . . . . 5 ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝐴𝑉𝐵𝑉))
1615adantl 481 . . . 4 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (𝐴𝑉𝐵𝑉))
1716adantl 481 . . 3 ((𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∧ ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))) → (𝐴𝑉𝐵𝑉))
18 wspthnon 26809 . . . . 5 (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
1918biimpi 206 . . . 4 (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
2019adantr 480 . . 3 ((𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∧ ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
2110, 17, 203jca 1261 . 2 ((𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∧ ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))
229, 21mpdan 703 1 (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∀wral 2941  {crab 2945  Vcvv 3231   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  ℕ0cn0 11330  Vtxcvtx 25919  SPathsOncspthson 26667   WWalksNOn cwwlksnon 26775   WSPathsNOn cwwspthsnon 26777 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-wwlksnon 26780  df-wspthsnon 26782 This theorem is referenced by:  wspthneq1eq2  26814  wspthsnonn0vne  26882  wspthsswwlknon  26886
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