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Theorem trlsegvdeglem5 30253
Description: Lemma for trlsegvdeg 30256. (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 5919 . 2 (𝜑 → dom (iEdg‘𝑌) = dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
3 fvex 6920 . . 3 (𝐼‘(𝐹𝑁)) ∈ V
4 dmsnopg 6235 . . 3 ((𝐼‘(𝐹𝑁)) ∈ V → dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {(𝐹𝑁)})
53, 4mp1i 13 . 2 (𝜑 → dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {(𝐹𝑁)})
62, 5eqtrd 2775 1 (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631  cop 4637   class class class wbr 5148  dom cdm 5689  cres 5691  cima 5692  Fun wfun 6557  cfv 6563  (class class class)co 7431  0cc0 11153  ...cfz 13544  ..^cfzo 13691  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  Trailsctrls 29723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571
This theorem is referenced by:  trlsegvdeglem7  30255  trlsegvdeg  30256
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