Theorem trlsegvdeglem2 30308

Description: Lemma for trlsegvdeg 30314. (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 6543	. 2 ⊢ (𝜑 → Fun (𝐼 ↾ (𝐹 “ (0..^𝑁))))
3		trlsegvdeg.ix	. . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
4	3	funeqd 6522	. 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 1542   ∈ wcel 2114  {csn 4582  ⟨cop 4588   class class class wbr 5100   ↾ cres 5634   “ cima 5635  Fun wfun 6494  ‘cfv 6500  (class class class)co 7368  0cc0 11038  ...cfz 13435  ..^cfzo 13582  ♯chash 14265  Vtxcvtx 29081  iEdgciedg 29082  Trailsctrls 29774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-ss 3920  df-br 5101  df-opab 5163  df-rel 5639  df-cnv 5640  df-co 5641  df-res 5644  df-fun 6502

This theorem is referenced by:  trlsegvdeg  30314