Theorem trlsonprop 28076

Description: Properties of a trail between two vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 16-Jan-2021.)

Hypothesis

Ref	Expression
trlsonfval.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
trlsonprop	⊢ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))

Proof of Theorem trlsonprop

Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trlsonfval.v	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	istrlson 28075	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))
3	2	3adantl1 1165	. 2 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))
4		df-trlson 28061	. 2 ⊢ TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Trails‘𝑔)𝑝)}))
5	1, 3, 4	wksonproplem 28072	1 ⊢ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Vtxcvtx 27366  WalksOncwlkson 27964  Trailsctrls 28058  TrailsOnctrlson 28059

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-trlson 28061

This theorem is referenced by:  trlsonistrl  28077  trlsonwlkon  28078