Theorem numclwwlk1lem2fv 30388

Description: Value of the function 𝑇. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Oct-2022.)

Hypotheses

Ref	Expression
extwwlkfab.v	⊢ 𝑉 = (Vtx‘𝐺)
extwwlkfab.c	⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
extwwlkfab.f	⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))
numclwwlk.t	⊢ 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))⟩)

Assertion

Ref	Expression
numclwwlk1lem2fv	⊢ (𝑊 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑊) = ⟨(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))⟩)

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

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

Proof of Theorem numclwwlk1lem2fv

Step	Hyp	Ref	Expression
1		oveq1 7455	. . 3 ⊢ (𝑢 = 𝑊 → (𝑢 prefix (𝑁 − 2)) = (𝑊 prefix (𝑁 − 2)))
2		fveq1 6919	. . 3 ⊢ (𝑢 = 𝑊 → (𝑢‘(𝑁 − 1)) = (𝑊‘(𝑁 − 1)))
3	1, 2	opeq12d 4905	. 2 ⊢ (𝑢 = 𝑊 → ⟨(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))⟩ = ⟨(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))⟩)
4		numclwwlk.t	. 2 ⊢ 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))⟩)
5		opex 5484	. 2 ⊢ ⟨(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))⟩ ∈ V
6	3, 4, 5	fvmpt 7029	1 ⊢ (𝑊 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑊) = ⟨(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))⟩)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {crab 3443  ⟨cop 4654   ↦ cmpt 5249  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1c1 11185   − cmin 11520  2c2 12348  ℤ_≥cuz 12903   prefix cpfx 14718  Vtxcvtx 29031  ClWWalksNOncclwwlknon 30119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451

This theorem is referenced by:  numclwwlk1lem2f1  30389  numclwwlk1lem2fo  30390