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Theorem trlsegvdeglem5 30243
Description: Lemma for trlsegvdeg 30246. (Contributed by AV, 21-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
trlsegvdeglem5 (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})

Proof of Theorem trlsegvdeglem5
StepHypRef Expression
1 trlsegvdeg.iy . . 3 (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
21dmeqd 5916 . 2 (𝜑 → dom (iEdg‘𝑌) = dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
3 fvex 6919 . . 3 (𝐼‘(𝐹𝑁)) ∈ V
4 dmsnopg 6233 . . 3 ((𝐼‘(𝐹𝑁)) ∈ V → dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {(𝐹𝑁)})
53, 4mp1i 13 . 2 (𝜑 → dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {(𝐹𝑁)})
62, 5eqtrd 2777 1 (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632   class class class wbr 5143  dom cdm 5685  cres 5687  cima 5688  Fun wfun 6555  cfv 6561  (class class class)co 7431  0cc0 11155  ...cfz 13547  ..^cfzo 13694  chash 14369  Vtxcvtx 29013  iEdgciedg 29014  Trailsctrls 29708
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-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
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-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569
This theorem is referenced by:  trlsegvdeglem7  30245  trlsegvdeg  30246
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