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Theorem trlsegvdeglem5 41392
 Description: Lemma for trlsegvdeg 41395. (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 5248 . 2 (𝜑 → dom (iEdg‘𝑌) = dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
3 fvex 6113 . . 3 (𝐼‘(𝐹𝑁)) ∈ V
4 dmsnopg 5524 . . 3 ((𝐼‘(𝐹𝑁)) ∈ V → dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {(𝐹𝑁)})
53, 4mp1i 13 . 2 (𝜑 → dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {(𝐹𝑁)})
62, 5eqtrd 2644 1 (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131   class class class wbr 4583  dom cdm 5038   ↾ cres 5040   “ cima 5041  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ...cfz 12197  ..^cfzo 12334  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674  TrailSctrls 40899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812 This theorem is referenced by:  trlsegvdeglem7  41394  trlsegvdeg  41395
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