Theorem av-numclwlk1lem2fv 41523

Description: Value of the function 𝑇. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.)

Hypotheses

Ref	Expression
av-extwwlkfab.v	⊢ 𝑉 = (Vtx‘𝐺)
av-extwwlkfab.f	⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣})
av-extwwlkfab.c	⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
av-numclwwlk.t	⊢ 𝑇 = (𝑤 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)

Assertion

Ref	Expression
av-numclwlk1lem2fv	⊢ (𝑃 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑃) = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩)

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

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

Proof of Theorem av-numclwlk1lem2fv

Step	Hyp	Ref	Expression
1		av-numclwwlk.t	. . 3 ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)
2	1	a1i 11	. 2 ⊢ (𝑃 ∈ (𝑋𝐶𝑁) → 𝑇 = (𝑤 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩))
3		oveq1 6556	. . . 4 ⊢ (𝑤 = 𝑃 → (𝑤 substr ⟨0, (𝑁 − 2)⟩) = (𝑃 substr ⟨0, (𝑁 − 2)⟩))
4		fveq1 6102	. . . 4 ⊢ (𝑤 = 𝑃 → (𝑤‘(𝑁 − 1)) = (𝑃‘(𝑁 − 1)))
5	3, 4	opeq12d 4348	. . 3 ⊢ (𝑤 = 𝑃 → ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩ = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩)
6	5	adantl 481	. 2 ⊢ ((𝑃 ∈ (𝑋𝐶𝑁) ∧ 𝑤 = 𝑃) → ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩ = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩)
7		id 22	. 2 ⊢ (𝑃 ∈ (𝑋𝐶𝑁) → 𝑃 ∈ (𝑋𝐶𝑁))
8		opex 4859	. . 3 ⊢ ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩ ∈ V
9	8	a1i 11	. 2 ⊢ (𝑃 ∈ (𝑋𝐶𝑁) → ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩ ∈ V)
10	2, 6, 7, 9	fvmptd 6197	1 ⊢ (𝑃 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑃) = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816   − cmin 10145  ℕcn 10897  2c2 10947  ℤ_≥cuz 11563   substr csubstr 13150  Vtxcvtx 25673   ClWWalkSN cclwwlksn 41184

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-csb 3500  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:  av-numclwlk1lem2f1  41524  av-numclwlk1lem2fo  41525