Theorem numclwwlk1lem2fv 30318

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 7360	. . 3 ⊢ (𝑢 = 𝑊 → (𝑢 prefix (𝑁 − 2)) = (𝑊 prefix (𝑁 − 2)))
2		fveq1 6825	. . 3 ⊢ (𝑢 = 𝑊 → (𝑢‘(𝑁 − 1)) = (𝑊‘(𝑁 − 1)))
3	1, 2	opeq12d 4835	. 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 5411	. 2 ⊢ ⟨(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))⟩ ∈ V
6	3, 4, 5	fvmpt 6934	1 ⊢ (𝑊 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑊) = ⟨(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3396  ⟨cop 4585   ↦ cmpt 5176  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  1c1 11029   − cmin 11365  2c2 12201  ℤ_≥cuz 12753   prefix cpfx 14595  Vtxcvtx 28959  ClWWalksNOncclwwlknon 30049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356

This theorem is referenced by:  numclwwlk1lem2f1  30319  numclwwlk1lem2fo  30320