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Theorem trlsegvdeglem2 28009
Description: Lemma for trlsegvdeg 28015. (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 6370 . . 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 6350 . 2 (𝜑 → (Fun (iEdg‘𝑋) ↔ Fun (𝐼 ↾ (𝐹 “ (0..^𝑁)))))
63, 5mpbird 260 1 (𝜑 → Fun (iEdg‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  {csn 4528  cop 4534   class class class wbr 5033  cres 5525  cima 5526  Fun wfun 6322  cfv 6328  (class class class)co 7139  0cc0 10530  ...cfz 12889  ..^cfzo 13032  chash 13690  Vtxcvtx 26792  iEdgciedg 26793  Trailsctrls 27483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901  df-br 5034  df-opab 5096  df-rel 5530  df-cnv 5531  df-co 5532  df-res 5535  df-fun 6330
This theorem is referenced by:  trlsegvdeg  28015
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