Theorem numclwlk2lem2fv 30866

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 7424	. . 3 ⊢ (𝑥 = 𝑊 → (𝑥 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1)))
3		simpr 490	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → 𝑊 ∈ (𝑋𝐻(𝑁 + 2)))
4		ovexd 7452	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑊 prefix (𝑁 + 1)) ∈ V)
5	1, 2, 3, 4	fvmptd3 7014	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑊 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑅‘𝑊) = (𝑊 prefix (𝑁 + 1)))
6	5	ex 418	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑊) = (𝑊 prefix (𝑁 + 1))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  {crab 3414  Vcvv 3453   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  0cc0 11128  1c1 11129   + caddc 11131   − cmin 11469  ℕcn 12261  2c2 12323  ℤ_≥cuz 12891  lastSclsw 14631   prefix cpfx 14744  Vtxcvtx 29461   WWalksN cwwlksn 30302  ClWWalksNOncclwwlknon 30565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420

This theorem is used by:  numclwlk2lem2f1o  30867