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Theorem numclwwlkovh 27106
Description: Value of operation 𝐻, mapping a vertex 𝑣 and a positive integer 𝑛 to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v" according to definition 7 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 26-Aug-2018.) (Revised by AV, 30-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))})
Assertion
Ref Expression
numclwwlkovh ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))})
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤   𝑤,𝑉
Allowed substitution hints:   𝑄(𝑤,𝑣,𝑛)   𝐹(𝑤,𝑣,𝑛)   𝐻(𝑤,𝑣,𝑛)

Proof of Theorem numclwwlkovh
StepHypRef Expression
1 oveq1 6617 . . . 4 (𝑛 = 𝑁 → (𝑛 ClWWalksN 𝐺) = (𝑁 ClWWalksN 𝐺))
21adantl 482 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → (𝑛 ClWWalksN 𝐺) = (𝑁 ClWWalksN 𝐺))
3 eqeq2 2632 . . . 4 (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
4 oveq1 6617 . . . . . 6 (𝑛 = 𝑁 → (𝑛 − 2) = (𝑁 − 2))
54fveq2d 6157 . . . . 5 (𝑛 = 𝑁 → (𝑤‘(𝑛 − 2)) = (𝑤‘(𝑁 − 2)))
65neeq1d 2849 . . . 4 (𝑛 = 𝑁 → ((𝑤‘(𝑛 − 2)) ≠ (𝑤‘0) ↔ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0)))
73, 6bi2anan9 916 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → (((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0)) ↔ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))))
82, 7rabeqbidv 3184 . 2 ((𝑣 = 𝑋𝑛 = 𝑁) → {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))} = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))})
9 numclwwlk.h . 2 𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})
10 ovex 6638 . . 3 (𝑁 ClWWalksN 𝐺) ∈ V
1110rabex 4778 . 2 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))} ∈ V
128, 9, 11ovmpt2a 6751 1 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  {crab 2911  cfv 5852  (class class class)co 6610  cmpt2 6612  0cc0 9888  cmin 10218  cn 10972  2c2 11022   lastS clsw 13239  Vtxcvtx 25791   WWalksN cwwlksn 26604   ClWWalksN cclwwlksn 26760
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  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-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615
This theorem is referenced by:  numclwwlk2lem1  27107  numclwlk2lem2f  27108  numclwlk2lem2f1o  27110  numclwwlk3lem  27113
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