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Theorem trlsegvdeglem3 26982
Description: Lemma for trlsegvdeg 26987. (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
trlsegvdeglem3 (𝜑 → Fun (iEdg‘𝑌))

Proof of Theorem trlsegvdeglem3
StepHypRef Expression
1 fvex 6168 . . . 4 (𝐹𝑁) ∈ V
2 fvex 6168 . . . 4 (𝐼‘(𝐹𝑁)) ∈ V
31, 2pm3.2i 471 . . 3 ((𝐹𝑁) ∈ V ∧ (𝐼‘(𝐹𝑁)) ∈ V)
4 funsng 5905 . . 3 (((𝐹𝑁) ∈ V ∧ (𝐼‘(𝐹𝑁)) ∈ V) → Fun {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
53, 4mp1i 13 . 2 (𝜑 → Fun {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
6 trlsegvdeg.iy . . 3 (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
76funeqd 5879 . 2 (𝜑 → (Fun (iEdg‘𝑌) ↔ Fun {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
85, 7mpbird 247 1 (𝜑 → Fun (iEdg‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  {csn 4155  cop 4161   class class class wbr 4623  cres 5086  cima 5087  Fun wfun 5851  cfv 5857  (class class class)co 6615  0cc0 9896  ...cfz 12284  ..^cfzo 12422  #chash 13073  Vtxcvtx 25808  iEdgciedg 25809  Trailsctrls 26490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-iota 5820  df-fun 5859  df-fv 5865
This theorem is referenced by:  trlsegvdeg  26987
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