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Theorem trlsonistrl 28486
Description: A trail between two vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 7-Jan-2021.)
Assertion
Ref Expression
trlsonistrl (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)

Proof of Theorem trlsonistrl
StepHypRef Expression
1 eqid 2738 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
21trlsonprop 28485 . 2 (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)))
3 simp3r 1203 . 2 (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)) → 𝐹(Trails‘𝐺)𝑃)
42, 3syl 17 1 (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  Vcvv 3444   class class class wbr 5104  cfv 6494  (class class class)co 7352  Vtxcvtx 27776  WalksOncwlkson 28374  Trailsctrls 28467  TrailsOnctrlson 28468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7914  df-2nd 7915  df-trlson 28470
This theorem is referenced by: (None)
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