Theorem trlsegvdeglem2 30150

Description: Lemma for trlsegvdeg 30156. (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 6559	. 2 ⊢ (𝜑 → Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))
3		trlsegvdeg.ix	. . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
4	3	funeqd 6538	. 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 4589  ⟨cop 4595   class class class wbr 5107   ↾ cres 5640   “ cima 5641  Fun wfun 6505  ‘cfv 6511  (class class class)co 7387  0cc0 11068  ...cfz 13468  ..^cfzo 13615  ♯chash 14295  Vtxcvtx 28923  iEdgciedg 28924  Trailsctrls 29618

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 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-in 3921  df-ss 3931  df-br 5108  df-opab 5170  df-rel 5645  df-cnv 5646  df-co 5647  df-res 5650  df-fun 6513

This theorem is referenced by:  trlsegvdeg  30156