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Theorem trlsegvdeglem3 29455
Description: Lemma for trlsegvdeg 29460. (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 6901 . . . 4 (𝐹𝑁) ∈ V
2 fvex 6901 . . . 4 (𝐼‘(𝐹𝑁)) ∈ V
31, 2pm3.2i 472 . . 3 ((𝐹𝑁) ∈ V ∧ (𝐼‘(𝐹𝑁)) ∈ V)
4 funsng 6596 . . 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 6567 . 2 (𝜑 → (Fun (iEdg‘𝑌) ↔ Fun {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
85, 7mpbird 257 1 (𝜑 → Fun (iEdg‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  {csn 4627  cop 4633   class class class wbr 5147  cres 5677  cima 5678  Fun wfun 6534  cfv 6540  (class class class)co 7404  0cc0 11106  ...cfz 13480  ..^cfzo 13623  chash 14286  Vtxcvtx 28236  iEdgciedg 28237  Trailsctrls 28927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-iota 6492  df-fun 6542  df-fv 6548
This theorem is referenced by:  trlsegvdeg  29460
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