MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istrlson Structured version   Visualization version   GIF version

Theorem istrlson 29739
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 29738 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(TrailsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝑓(Trails‘𝐺)𝑝)})
32breqd 5158 . 2 ((𝐴𝑉𝐵𝑉) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝑓(Trails‘𝐺)𝑝)}𝑃))
4 breq12 5152 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃))
5 breq12 5152 . . . 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 2734 . . 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 5543 . 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 1536  wcel 2105   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:  trlsonprop  29740  trlontrl  29743  isspthonpth  29781  spthonepeq  29784  2trlond  29968  0trlon  30152  1pthond  30172  3trlond  30201
  Copyright terms: Public domain W3C validator