Theorem numclwwlk1lem2fv 30443

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 7375	. . 3 ⊢ (𝑢 = 𝑊 → (𝑢 prefix (𝑁 − 2)) = (𝑊 prefix (𝑁 − 2)))
2		fveq1 6841	. . 3 ⊢ (𝑢 = 𝑊 → (𝑢‘(𝑁 − 1)) = (𝑊‘(𝑁 − 1)))
3	1, 2	opeq12d 4839	. 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 5419	. 2 ⊢ ⟨(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))⟩ ∈ V
6	3, 4, 5	fvmpt 6949	1 ⊢ (𝑊 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑊) = ⟨(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))⟩)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3401  ⟨cop 4588   ↦ cmpt 5181  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1c1 11039   − cmin 11376  2c2 12212  ℤ_≥cuz 12763   prefix cpfx 14606  Vtxcvtx 29081  ClWWalksNOncclwwlknon 30174

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371

This theorem is referenced by:  numclwwlk1lem2f1  30444  numclwwlk1lem2fo  30445