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Theorem istrlson 29725
Description: Properties of a pair of functions to be a trail between two given vertices. (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 21-Mar-2021.)
Hypothesis
Ref Expression
trlsonfval.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
istrlson (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)))

Proof of Theorem istrlson
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trlsonfval.v . . . 4 𝑉 = (Vtx‘𝐺)
21trlsonfval 29724 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(TrailsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝑓(Trails‘𝐺)𝑝)})
32breqd 5154 . 2 ((𝐴𝑉𝐵𝑉) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝑓(Trails‘𝐺)𝑝)}𝑃))
4 breq12 5148 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃))
5 breq12 5148 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓(Trails‘𝐺)𝑝𝐹(Trails‘𝐺)𝑃))
64, 5anbi12d 632 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝑓(Trails‘𝐺)𝑝) ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)))
7 eqid 2737 . . 3 {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝑓(Trails‘𝐺)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝑓(Trails‘𝐺)𝑝)}
86, 7brabga 5539 . 2 ((𝐹𝑈𝑃𝑍) → (𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝑓(Trails‘𝐺)𝑝)}𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)))
93, 8sylan9bb 509 1 (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143  {copab 5205  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  WalksOncwlkson 29615  Trailsctrls 29708  TrailsOnctrlson 29709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-trlson 29711
This theorem is referenced by:  trlsonprop  29726  trlontrl  29729  isspthonpth  29769  spthonepeq  29772  2trlond  29959  0trlon  30143  1pthond  30163  3trlond  30192
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