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Theorem numclwlk2lem2fv 27086
 Description: Value of the function R. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.)
Hypotheses
Ref Expression
numclwwlk.v 𝑉 = (Vtx‘𝐺)
numclwwlk.q 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
numclwwlk.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
numclwwlk.h 𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})
numclwwlk.r 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))
Assertion
Ref Expression
numclwlk2lem2fv ((𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅𝑊) = (𝑊 substr ⟨0, (𝑁 + 1)⟩)))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤   𝑤,𝑉   𝑣,𝑊,𝑤   𝑥,𝐺,𝑤   𝑥,𝐻   𝑥,𝑁   𝑥,𝑄   𝑥,𝑉   𝑥,𝑋   𝑥,𝑊
Allowed substitution hints:   𝑄(𝑤,𝑣,𝑛)   𝑅(𝑥,𝑤,𝑣,𝑛)   𝐹(𝑥,𝑤,𝑣,𝑛)   𝐻(𝑤,𝑣,𝑛)   𝑊(𝑛)

Proof of Theorem numclwlk2lem2fv
StepHypRef Expression
1 numclwwlk.r . . . 4 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))
21a1i 11 . . 3 (((𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩)))
3 oveq1 6612 . . . 4 (𝑥 = 𝑊 → (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑊 substr ⟨0, (𝑁 + 1)⟩))
43adantl 482 . . 3 ((((𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 = 𝑊) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑊 substr ⟨0, (𝑁 + 1)⟩))
5 simpr 477 . . 3 (((𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → 𝑊 ∈ (𝑋𝐻(𝑁 + 2)))
6 ovex 6633 . . . 4 (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ V
76a1i 11 . . 3 (((𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ V)
82, 4, 5, 7fvmptd 6246 . 2 (((𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑅𝑊) = (𝑊 substr ⟨0, (𝑁 + 1)⟩))
98ex 450 1 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅𝑊) = (𝑊 substr ⟨0, (𝑁 + 1)⟩)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1992   ≠ wne 2796  {crab 2916  Vcvv 3191  ⟨cop 4159   ↦ cmpt 4678  ‘cfv 5850  (class class class)co 6605   ↦ cmpt2 6607  0cc0 9881  1c1 9882   + caddc 9884   − cmin 10211  ℕcn 10965  2c2 11015   lastS clsw 13226   substr csubstr 13229  Vtxcvtx 25769   WWalksN cwwlksn 26581   ClWWalksN cclwwlksn 26737 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608 This theorem is referenced by:  numclwlk2lem2f1o  27087
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