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Theorem trlsonfval 29738
Description: The set of trails between two vertices. (Contributed by Alexander van der Vekens, 4-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 15-Jan-2021.) (Revised by AV, 21-Mar-2021.)
Hypothesis
Ref Expression
trlsonfval.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
trlsonfval ((𝐴𝑉𝐵𝑉) → (𝐴(TrailsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝑓(Trails‘𝐺)𝑝)})
Distinct variable groups:   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝑓,𝐺,𝑝   𝑓,𝑉,𝑝

Proof of Theorem trlsonfval
Dummy variables 𝑎 𝑏 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trlsonfval.v . . . 4 𝑉 = (Vtx‘𝐺)
211vgrex 29033 . . 3 (𝐴𝑉𝐺 ∈ V)
32adantr 480 . 2 ((𝐴𝑉𝐵𝑉) → 𝐺 ∈ V)
4 simpl 482 . . 3 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
54, 1eleqtrdi 2848 . 2 ((𝐴𝑉𝐵𝑉) → 𝐴 ∈ (Vtx‘𝐺))
6 simpr 484 . . 3 ((𝐴𝑉𝐵𝑉) → 𝐵𝑉)
76, 1eleqtrdi 2848 . 2 ((𝐴𝑉𝐵𝑉) → 𝐵 ∈ (Vtx‘𝐺))
8 df-trlson 29725 . 2 TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝𝑓(Trails‘𝑔)𝑝)}))
93, 5, 7, 8mptmpoopabovd 8105 1 ((𝐴𝑉𝐵𝑉) → (𝐴(TrailsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝑓(Trails‘𝐺)𝑝)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  Vcvv 3477   class class class wbr 5147  {copab 5209  cfv 6562  (class class class)co 7430  Vtxcvtx 29027  WalksOncwlkson 29629  Trailsctrls 29722  TrailsOnctrlson 29723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-trlson 29725
This theorem is referenced by:  istrlson  29739
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