Theorem trlsegvdeglem2 16311

Description: Lemma for trlsegvdeg . (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	⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
trlsegvdeg.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
trlsegvdeg.w	⊢ (𝜑 → 𝐹(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) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
trlsegvdeg.iz	⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))

Assertion

Ref	Expression
trlsegvdeglem2	⊢ (𝜑 → Fun (iEdg‘𝑋))

Proof of Theorem trlsegvdeglem2

Step	Hyp	Ref	Expression
1		trlsegvdeg.f	. . 3 ⊢ (𝜑 → Fun 𝐼)
2	1	funresd 5368	. 2 ⊢ (𝜑 → Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))
3		trlsegvdeg.ix	. . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
4	3	funeqd 5348	. 2 ⊢ (𝜑 → (Fun (iEdg‘𝑋) ↔ Fun (𝐼 ↾ (𝐹 “ (0..^𝑁)))))
5	2, 4	mpbird 167	1 ⊢ (𝜑 → Fun (iEdg‘𝑋))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  {csn 3669  ⟨cop 3672   class class class wbr 4088   ↾ cres 4727   “ cima 4728  Fun wfun 5320  ‘cfv 5326  (class class class)co 6017  0cc0 8031  ...cfz 10242  ..^cfzo 10376  ♯chash 11036  Vtxcvtx 15862  iEdgciedg 15863  Trailsctrls 16230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-br 4089  df-opab 4151  df-rel 4732  df-cnv 4733  df-co 4734  df-res 4737  df-fun 5328

This theorem is referenced by: (None)