Theorem trlsegvdeglem2 30513

Description: Lemma for trlsegvdeg 30519. (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 6580	. 2 ⊢ (𝜑 → Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))
3		trlsegvdeg.ix	. . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
4	3	funeqd 6559	. 2 ⊢ (𝜑 → (Fun (iEdg‘𝑋) ↔ Fun (𝐼 ↾ (𝐹 “ (0..^𝑁)))))
5	2, 4	mpbird 260	1 ⊢ (𝜑 → Fun (iEdg‘𝑋))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {csn 4594  ⟨cop 4600   class class class wbr 5113   ↾ cres 5664   “ cima 5665  Fun wfun 6531  ‘cfv 6537  (class class class)co 7411  0cc0 11100  ...cfz 13535  ..^cfzo 13682  ♯chash 14366  Vtxcvtx 29287  iEdgciedg 29288  Trailsctrls 29979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-in 3920  df-ss 3930  df-br 5114  df-opab 5178  df-rel 5669  df-cnv 5670  df-co 5671  df-res 5674  df-fun 6539

This theorem is referenced by:  trlsegvdeg  30519