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Theorem wspthnonp 29380
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 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
wspthnonp (π‘Š ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) β†’ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (π‘Š ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)π‘Š)))
Distinct variable groups:   𝐴,𝑓   𝐡,𝑓   𝑓,𝐺   𝑓,𝑁   𝑓,𝑉   𝑓,π‘Š
Proof of Theorem wspthnonp
Dummy variables 𝑀 π‘Ž 𝑏 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6903 . . . . 5 (Vtxβ€˜π‘”) ∈ V
21, 1pm3.2i 469 . . . 4 ((Vtxβ€˜π‘”) ∈ V ∧ (Vtxβ€˜π‘”) ∈ V)
32rgen2w 3064 . . 3 βˆ€π‘› ∈ β„•0 βˆ€π‘” ∈ V ((Vtxβ€˜π‘”) ∈ V ∧ (Vtxβ€˜π‘”) ∈ V)
4 df-wspthsnon 29355 . . . 4 WSPathsNOn = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀}))
5 fveq2 6890 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
65, 5jca 510 . . . . 5 (𝑔 = 𝐺 β†’ ((Vtxβ€˜π‘”) = (Vtxβ€˜πΊ) ∧ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ)))
76adantl 480 . . . 4 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ ((Vtxβ€˜π‘”) = (Vtxβ€˜πΊ) ∧ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ)))
84, 7el2mpocl 8074 . . 3 (βˆ€π‘› ∈ β„•0 βˆ€π‘” ∈ V ((Vtxβ€˜π‘”) ∈ V ∧ (Vtxβ€˜π‘”) ∈ V) β†’ (π‘Š ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) β†’ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)))))
93, 8ax-mp 5 . 2 (π‘Š ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) β†’ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ))))
10 simprl 767 . . 3 ((π‘Š ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) ∧ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)))) β†’ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V))
11 wspthnonp.v . . . . . . . 8 𝑉 = (Vtxβ€˜πΊ)
1211eleq2i 2823 . . . . . . 7 (𝐴 ∈ 𝑉 ↔ 𝐴 ∈ (Vtxβ€˜πΊ))
1311eleq2i 2823 . . . . . . 7 (𝐡 ∈ 𝑉 ↔ 𝐡 ∈ (Vtxβ€˜πΊ))
1412, 13anbi12i 625 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ↔ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)))
1514biimpri 227 . . . . 5 ((𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉))
1615adantl 480 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ))) β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉))
1716adantl 480 . . 3 ((π‘Š ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) ∧ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)))) β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉))
18 wspthnon 29379 . . . . 5 (π‘Š ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) ↔ (π‘Š ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)π‘Š))
1918biimpi 215 . . . 4 (π‘Š ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) β†’ (π‘Š ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)π‘Š))
2019adantr 479 . . 3 ((π‘Š ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) ∧ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)))) β†’ (π‘Š ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)π‘Š))
2110, 17, 203jca 1126 . 2 ((π‘Š ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) ∧ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)))) β†’ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (π‘Š ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)π‘Š)))
229, 21mpdan 683 1 (π‘Š ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐡) β†’ ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (π‘Š ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) ∧ βˆƒπ‘“ 𝑓(𝐴(SPathsOnβ€˜πΊ)𝐡)π‘Š)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  βˆ€wral 3059  {crab 3430  Vcvv 3472   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  β„•0cn0 12476  Vtxcvtx 28523  SPathsOncspthson 29239   WWalksNOn cwwlksnon 29348   WSPathsNOn cwwspthsnon 29350
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-wwlksnon 29353  df-wspthsnon 29355
This theorem is referenced by:  wspthneq1eq2  29381  wspthsnonn0vne  29438  wspthsswwlknon  29442
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