Theorem numclwlk1lem2fv 27346

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

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 substr ⟨0, (𝑁 − 2)⟩), (𝑢‘(𝑁 − 1))⟩)

Assertion

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

Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑉,𝑣,𝑤   𝑛,𝑋,𝑣,𝑤   𝑛,𝑁,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 numclwlk1lem2fv

Step	Hyp	Ref	Expression
1		oveq1 6697	. . 3 ⊢ (𝑢 = 𝑊 → (𝑢 substr ⟨0, (𝑁 − 2)⟩) = (𝑊 substr ⟨0, (𝑁 − 2)⟩))
2		fveq1 6228	. . 3 ⊢ (𝑢 = 𝑊 → (𝑢‘(𝑁 − 1)) = (𝑊‘(𝑁 − 1)))
3	1, 2	opeq12d 4441	. 2 ⊢ (𝑢 = 𝑊 → ⟨(𝑢 substr ⟨0, (𝑁 − 2)⟩), (𝑢‘(𝑁 − 1))⟩ = ⟨(𝑊 substr ⟨0, (𝑁 − 2)⟩), (𝑊‘(𝑁 − 1))⟩)
4		numclwwlk.t	. 2 ⊢ 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑢 substr ⟨0, (𝑁 − 2)⟩), (𝑢‘(𝑁 − 1))⟩)
5		opex 4962	. 2 ⊢ ⟨(𝑊 substr ⟨0, (𝑁 − 2)⟩), (𝑊‘(𝑁 − 1))⟩ ∈ V
6	3, 4, 5	fvmpt 6321	1 ⊢ (𝑊 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑊) = ⟨(𝑊 substr ⟨0, (𝑁 − 2)⟩), (𝑊‘(𝑁 − 1))⟩)

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  {crab 2945  ⟨cop 4216   ↦ cmpt 4762  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  0cc0 9974  1c1 9975   − cmin 10304  2c2 11108  ℤ_≥cuz 11725   substr csubstr 13327  Vtxcvtx 25919  ClWWalksNOncclwwlknon 27060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693

This theorem is referenced by:  numclwlk1lem2f1  27347  numclwlk1lem2fo  27348