Theorem trlsegvdeglem2 30156

Description: Lemma for trlsegvdeg 30162. (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 6561	. 2 ⊢ (𝜑 → Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))
3		trlsegvdeg.ix	. . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
4	3	funeqd 6540	. 2 ⊢ (𝜑 → (Fun (iEdg‘𝑋) ↔ Fun (𝐼 ↾ (𝐹 “ (0..^𝑁)))))
5	2, 4	mpbird 257	1 ⊢ (𝜑 → Fun (iEdg‘𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4591  ⟨cop 4597   class class class wbr 5109   ↾ cres 5642   “ cima 5643  Fun wfun 6507  ‘cfv 6513  (class class class)co 7389  0cc0 11074  ...cfz 13474  ..^cfzo 13621  ♯chash 14301  Vtxcvtx 28929  iEdgciedg 28930  Trailsctrls 29624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3923  df-ss 3933  df-br 5110  df-opab 5172  df-rel 5647  df-cnv 5648  df-co 5649  df-res 5652  df-fun 6515

This theorem is referenced by:  trlsegvdeg  30162