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

Theorem trlsegvdeglem2 27927
Description: Lemma for trlsegvdeg 27933. (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 6390 . . 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 6370 . 2 (𝜑 → (Fun (iEdg‘𝑋) ↔ Fun (𝐼 ↾ (𝐹 “ (0..^𝑁)))))
63, 5mpbird 258 1 (𝜑 → Fun (iEdg‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {csn 4557  cop 4563   class class class wbr 5057  cres 5550  cima 5551  Fun wfun 6342  cfv 6348  (class class class)co 7145  0cc0 10525  ...cfz 12880  ..^cfzo 13021  chash 13678  Vtxcvtx 26708  iEdgciedg 26709  Trailsctrls 27399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-ss 3949  df-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-res 5560  df-fun 6350
This theorem is referenced by:  trlsegvdeg  27933
  Copyright terms: Public domain W3C validator