Theorem numclwlk1lem2fv 27226

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.f	⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
extwwlkfab.c	⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
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,𝑢,𝑣,𝑤   𝑛,𝑁,𝑢,𝑣,𝑤   𝑛,𝑉,𝑣,𝑤   𝑛,𝑋,𝑢,𝑣,𝑤   𝑤,𝐹   𝑤,𝑊   𝑢,𝐶   𝑢,𝐹   𝑢,𝑉   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 6657	. . 3 ⊢ (𝑢 = 𝑊 → (𝑢 substr ⟨0, (𝑁 − 2)⟩) = (𝑊 substr ⟨0, (𝑁 − 2)⟩))
2		fveq1 6190	. . 3 ⊢ (𝑢 = 𝑊 → (𝑢‘(𝑁 − 1)) = (𝑊‘(𝑁 − 1)))
3	1, 2	opeq12d 4410	. 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 4932	. 2 ⊢ ⟨(𝑊 substr ⟨0, (𝑁 − 2)⟩), (𝑊‘(𝑁 − 1))⟩ ∈ V
6	3, 4, 5	fvmpt 6282	1 ⊢ (𝑊 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑊) = ⟨(𝑊 substr ⟨0, (𝑁 − 2)⟩), (𝑊‘(𝑁 − 1))⟩)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  ⟨cop 4183   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  0cc0 9936  1c1 9937   − cmin 10266  ℕcn 11020  2c2 11070  ℤ_≥cuz 11687   substr csubstr 13295  Vtxcvtx 25874   ClWWalksN cclwwlksn 26876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  numclwlk1lem2f1  27227  numclwlk1lem2fo  27228