Theorem numclwlk2lem2fv 30473

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 7370	. . 3 ⊢ (𝑥 = 𝑊 → (𝑥 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1)))
3		simpr 485	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → 𝑊 ∈ (𝑋𝐻(𝑁 + 2)))
4		ovexd 7398	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑊 prefix (𝑁 + 1)) ∈ V)
5	1, 2, 3, 4	fvmptd3 6966	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑅‘𝑊) = (𝑊 prefix (𝑁 + 1)))
6	5	ex 413	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑊) = (𝑊 prefix (𝑁 + 1))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  {crab 3392  Vcvv 3432   ↦ cmpt 5160  ‘cfv 6492  (class class class)co 7363   ∈ cmpo 7365  0cc0 11036  1c1 11037   + caddc 11039   − cmin 11375  ℕcn 12172  2c2 12234  ℤ_≥cuz 12786  lastSclsw 14522   prefix cpfx 14631  Vtxcvtx 29090   WWalksN cwwlksn 29919  ClWWalksNOncclwwlknon 30182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366

This theorem is referenced by:  numclwlk2lem2f1o  30474