Theorem numclwlk2lem2fv 30353

Description: Value of the function 𝑅. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Revised by AV, 1-Nov-2022.)

Hypotheses

Ref	Expression
numclwwlk.v	⊢ 𝑉 = (Vtx‘𝐺)
numclwwlk.q	⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
numclwwlk.h	⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
numclwwlk.r	⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1)))

Assertion

Ref	Expression
numclwlk2lem2fv	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑊) = (𝑊 prefix (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)   𝐻(𝑤,𝑣,𝑛)   𝑊(𝑛)

Proof of Theorem numclwlk2lem2fv

Step	Hyp	Ref	Expression
1		numclwwlk.r	. . 3 ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1)))
2		oveq1 7353	. . 3 ⊢ (𝑥 = 𝑊 → (𝑥 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1)))
3		simpr 484	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → 𝑊 ∈ (𝑋𝐻(𝑁 + 2)))
4		ovexd 7381	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑊 prefix (𝑁 + 1)) ∈ V)
5	1, 2, 3, 4	fvmptd3 6952	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑅‘𝑊) = (𝑊 prefix (𝑁 + 1)))
6	5	ex 412	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑊) = (𝑊 prefix (𝑁 + 1))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  {crab 3395  Vcvv 3436   ↦ cmpt 5172  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  0cc0 11003  1c1 11004   + caddc 11006   − cmin 11341  ℕcn 12122  2c2 12177  ℤ_≥cuz 12729  lastSclsw 14466   prefix cpfx 14575  Vtxcvtx 28972   WWalksN cwwlksn 29802  ClWWalksNOncclwwlknon 30062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349

This theorem is referenced by:  numclwlk2lem2f1o  30354