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Theorem trlsonfval 28652
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 27951 . . 3 (𝐴𝑉𝐺 ∈ V)
32adantr 481 . 2 ((𝐴𝑉𝐵𝑉) → 𝐺 ∈ V)
4 simpl 483 . . 3 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
54, 1eleqtrdi 2848 . 2 ((𝐴𝑉𝐵𝑉) → 𝐴 ∈ (Vtx‘𝐺))
6 simpr 485 . . 3 ((𝐴𝑉𝐵𝑉) → 𝐵𝑉)
76, 1eleqtrdi 2848 . 2 ((𝐴𝑉𝐵𝑉) → 𝐵 ∈ (Vtx‘𝐺))
8 df-trlson 28639 . 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 8013 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 396   = wceq 1541  wcel 2106  Vcvv 3445   class class class wbr 5105  {copab 5167  cfv 6496  (class class class)co 7356  Vtxcvtx 27945  WalksOncwlkson 28543  Trailsctrls 28636  TrailsOnctrlson 28637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7920  df-2nd 7921  df-trlson 28639
This theorem is referenced by:  istrlson  28653
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