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Theorem wspthnonp 29381
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 6904 . . . . 5 (Vtxβ€˜π‘”) ∈ V
21, 1pm3.2i 470 . . . 4 ((Vtxβ€˜π‘”) ∈ V ∧ (Vtxβ€˜π‘”) ∈ V)
32rgen2w 3065 . . 3 βˆ€π‘› ∈ β„•0 βˆ€π‘” ∈ V ((Vtxβ€˜π‘”) ∈ V ∧ (Vtxβ€˜π‘”) ∈ V)
4 df-wspthsnon 29356 . . . 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 6891 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
65, 5jca 511 . . . . 5 (𝑔 = 𝐺 β†’ ((Vtxβ€˜π‘”) = (Vtxβ€˜πΊ) ∧ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ)))
76adantl 481 . . . 4 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ ((Vtxβ€˜π‘”) = (Vtxβ€˜πΊ) ∧ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ)))
84, 7el2mpocl 8076 . . 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 768 . . 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 2824 . . . . . . 7 (𝐴 ∈ 𝑉 ↔ 𝐴 ∈ (Vtxβ€˜πΊ))
1311eleq2i 2824 . . . . . . 7 (𝐡 ∈ 𝑉 ↔ 𝐡 ∈ (Vtxβ€˜πΊ))
1412, 13anbi12i 626 . . . . . 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 481 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ))) β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉))
1716adantl 481 . . 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 29380 . . . . 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 480 . . 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 1127 . 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 684 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 395   ∧ w3a 1086   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105  βˆ€wral 3060  {crab 3431  Vcvv 3473   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  β„•0cn0 12477  Vtxcvtx 28524  SPathsOncspthson 29240   WWalksNOn cwwlksnon 29349   WSPathsNOn cwwspthsnon 29351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-wwlksnon 29354  df-wspthsnon 29356
This theorem is referenced by:  wspthneq1eq2  29382  wspthsnonn0vne  29439  wspthsswwlknon  29443
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