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Theorem trlsegvdeglem5 27397
Description: Lemma for trlsegvdeg 27400. (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 5481 . 2 (𝜑 → dom (iEdg‘𝑌) = dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
3 fvex 6363 . . 3 (𝐼‘(𝐹𝑁)) ∈ V
4 dmsnopg 5765 . . 3 ((𝐼‘(𝐹𝑁)) ∈ V → dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {(𝐹𝑁)})
53, 4mp1i 13 . 2 (𝜑 → dom {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩} = {(𝐹𝑁)})
62, 5eqtrd 2794 1 (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321  cop 4327   class class class wbr 4804  dom cdm 5266  cres 5268  cima 5269  Fun wfun 6043  cfv 6049  (class class class)co 6814  0cc0 10148  ...cfz 12539  ..^cfzo 12679  chash 13331  Vtxcvtx 26094  iEdgciedg 26095  Trailsctrls 26818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-dm 5276  df-iota 6012  df-fv 6057
This theorem is referenced by:  trlsegvdeglem7  27399  trlsegvdeg  27400
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