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Theorem 2clwwlk 27504
 Description: Value of operation 𝐶, mapping a vertex v and an integer n greater than 1 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 6 in [Huneke] p. 2. Such closed walks are "double loops" consisting of a closed (n-2)-walk v = v(0) ... v(n-2) = v and a closed 2-walk v = v(n-2) v(n-1) v(n) = v. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 20-Apr-2022.)
Hypothesis
Ref Expression
2clwwlk.c 𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
Assertion
Ref Expression
2clwwlk ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤
Allowed substitution hints:   𝐶(𝑤,𝑣,𝑛)   𝑉(𝑤)

Proof of Theorem 2clwwlk
StepHypRef Expression
1 oveq12 6822 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → (𝑣(ClWWalksNOn‘𝐺)𝑛) = (𝑋(ClWWalksNOn‘𝐺)𝑁))
2 oveq1 6820 . . . . . 6 (𝑛 = 𝑁 → (𝑛 − 2) = (𝑁 − 2))
32fveq2d 6356 . . . . 5 (𝑛 = 𝑁 → (𝑤‘(𝑛 − 2)) = (𝑤‘(𝑁 − 2)))
43adantl 473 . . . 4 ((𝑣 = 𝑋𝑛 = 𝑁) → (𝑤‘(𝑛 − 2)) = (𝑤‘(𝑁 − 2)))
5 simpl 474 . . . 4 ((𝑣 = 𝑋𝑛 = 𝑁) → 𝑣 = 𝑋)
64, 5eqeq12d 2775 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → ((𝑤‘(𝑛 − 2)) = 𝑣 ↔ (𝑤‘(𝑁 − 2)) = 𝑋))
71, 6rabeqbidv 3335 . 2 ((𝑣 = 𝑋𝑛 = 𝑁) → {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣} = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})
8 2clwwlk.c . 2 𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
9 ovex 6841 . . 3 (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V
109rabex 4964 . 2 {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ∈ V
117, 8, 10ovmpt2a 6956 1 ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  {crab 3054  ‘cfv 6049  (class class class)co 6813   ↦ cmpt2 6815   − cmin 10458  2c2 11262  ℤ≥cuz 11879  ClWWalksNOncclwwlknon 27232 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818 This theorem is referenced by:  2clwwlk2  27505  2clwwlkel  27506  extwwlkfab  27511  numclwwlk3lemlem  27551  numclwwlk3lem  27552
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