Theorem numclwlk1lem2fv 26620

Description: Value of the function T. (Contributed by Alexander van der Vekens, 20-Sep-2018.)

Hypotheses

Ref	Expression
numclwwlk.c	⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
numclwwlk.f	⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
numclwwlk.g	⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
numclwwlk.t	⊢ 𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)

Assertion

Ref	Expression
numclwlk1lem2fv	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑃 ∈ (𝑋𝐺𝑁) → (𝑇‘𝑃) = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩))

Distinct variable groups:   𝑛,𝐸   𝑛,𝑁   𝑛,𝑉   𝑤,𝐶   𝑤,𝑁   𝐶,𝑛,𝑣,𝑤   𝑣,𝑁   𝑛,𝑋,𝑣,𝑤   𝑣,𝑉   𝑤,𝐸   𝑤,𝑉   𝑤,𝐹   𝑤,𝑃   𝑤,𝐺

Allowed substitution hints:   𝑃(𝑣,𝑛)   𝑇(𝑤,𝑣,𝑛)   𝐸(𝑣)   𝐹(𝑣,𝑛)   𝐺(𝑣,𝑛)

Proof of Theorem numclwlk1lem2fv

Step	Hyp	Ref	Expression
1		opex 4859	. . . . . 6 ⊢ ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩ ∈ V
2	1	a1i 11	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩ ∈ V)
3	2	anim1i 590	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ 𝑃 ∈ (𝑋𝐺𝑁)) → (⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩ ∈ V ∧ 𝑃 ∈ (𝑋𝐺𝑁)))
4	3	ancomd 466	. . 3 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ 𝑃 ∈ (𝑋𝐺𝑁)) → (𝑃 ∈ (𝑋𝐺𝑁) ∧ ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩ ∈ V))
5		oveq1 6556	. . . . 5 ⊢ (𝑤 = 𝑃 → (𝑤 substr ⟨0, (𝑁 − 2)⟩) = (𝑃 substr ⟨0, (𝑁 − 2)⟩))
6		fveq1 6102	. . . . 5 ⊢ (𝑤 = 𝑃 → (𝑤‘(𝑁 − 1)) = (𝑃‘(𝑁 − 1)))
7	5, 6	opeq12d 4348	. . . 4 ⊢ (𝑤 = 𝑃 → ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩ = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩)
8		numclwwlk.t	. . . 4 ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)
9	7, 8	fvmptg 6189	. . 3 ⊢ ((𝑃 ∈ (𝑋𝐺𝑁) ∧ ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩ ∈ V) → (𝑇‘𝑃) = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩)
10	4, 9	syl 17	. 2 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ 𝑃 ∈ (𝑋𝐺𝑁)) → (𝑇‘𝑃) = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩)
11	10	ex 449	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑃 ∈ (𝑋𝐺𝑁) → (𝑇‘𝑃) = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816   − cmin 10145  2c2 10947  3c3 10948  ℕ₀cn0 11169  ℤ_≥cuz 11563   substr csubstr 13150   USGrph cusg 25859   ClWWalksN cclwwlkn 26277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552

This theorem is referenced by:  numclwlk1lem2f1  26621  numclwlk1lem2fo  26622