MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trlsegvdeglem2 Structured version   Visualization version   GIF version

Theorem trlsegvdeglem2 27074
Description: Lemma for trlsegvdeg 27080. (Contributed by AV, 20-Feb-2021.)
Hypotheses
Ref Expression
trlsegvdeg.v 𝑉 = (Vtx‘𝐺)
trlsegvdeg.i 𝐼 = (iEdg‘𝐺)
trlsegvdeg.f (𝜑 → Fun 𝐼)
trlsegvdeg.n (𝜑𝑁 ∈ (0..^(#‘𝐹)))
trlsegvdeg.u (𝜑𝑈𝑉)
trlsegvdeg.w (𝜑𝐹(Trails‘𝐺)𝑃)
trlsegvdeg.vx (𝜑 → (Vtx‘𝑋) = 𝑉)
trlsegvdeg.vy (𝜑 → (Vtx‘𝑌) = 𝑉)
trlsegvdeg.vz (𝜑 → (Vtx‘𝑍) = 𝑉)
trlsegvdeg.ix (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
trlsegvdeg.iy (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
trlsegvdeg.iz (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
Assertion
Ref Expression
trlsegvdeglem2 (𝜑 → Fun (iEdg‘𝑋))

Proof of Theorem trlsegvdeglem2
StepHypRef Expression
1 trlsegvdeg.f . . 3 (𝜑 → Fun 𝐼)
2 funres 5927 . . 3 (Fun 𝐼 → Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))
31, 2syl 17 . 2 (𝜑 → Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))
4 trlsegvdeg.ix . . 3 (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
54funeqd 5908 . 2 (𝜑 → (Fun (iEdg‘𝑋) ↔ Fun (𝐼 ↾ (𝐹 “ (0..^𝑁)))))
63, 5mpbird 247 1 (𝜑 → Fun (iEdg‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  {csn 4175  cop 4181   class class class wbr 4651  cres 5114  cima 5115  Fun wfun 5880  cfv 5886  (class class class)co 6647  0cc0 9933  ...cfz 12323  ..^cfzo 12461  #chash 13112  Vtxcvtx 25868  iEdgciedg 25869  Trailsctrls 26581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200  df-in 3579  df-ss 3586  df-br 4652  df-opab 4711  df-rel 5119  df-cnv 5120  df-co 5121  df-res 5124  df-fun 5888
This theorem is referenced by:  trlsegvdeg  27080
  Copyright terms: Public domain W3C validator