Theorem trlsegvdeglem4 30020

Description: Lemma for trlsegvdeg 30024. (Contributed by AV, 21-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
trlsegvdeglem4	⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))

Proof of Theorem trlsegvdeglem4

Step	Hyp	Ref	Expression
1		trlsegvdeg.ix	. . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
2	1	dmeqd 5902	. 2 ⊢ (𝜑 → dom (iEdg‘𝑋) = dom (𝐼 ↾ (𝐹 “ (0..^𝑁))))
3		dmres 6001	. 2 ⊢ dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)
4	2, 3	eqtrdi 2783	1 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099   ∩ cin 3943  {csn 4624  ⟨cop 4630   class class class wbr 5142  dom cdm 5672   ↾ cres 5674   “ cima 5675  Fun wfun 6536  ‘cfv 6542  (class class class)co 7414  0cc0 11130  ...cfz 13508  ..^cfzo 13651  ♯chash 14313  Vtxcvtx 28796  iEdgciedg 28797  Trailsctrls 29491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-dm 5682  df-res 5684

This theorem is referenced by:  trlsegvdeglem6  30022  trlsegvdeg  30024