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Theorem numclwwlkovhOLD 27573
Description: Obsolete version of numclwwlkovh0 27563 as of 1-May-2022. (Contributed by Alexander van der Vekens, 26-Aug-2018.) (Revised by AV, 30-May-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
numclwwlkOLD.v 𝑉 = (Vtx‘𝐺)
numclwwlkOLD.q 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
numclwwlkOLD.h 𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})
Assertion
Ref Expression
numclwwlkovhOLD ((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
Allowed substitution hints:   𝑄(𝑤,𝑣,𝑛)   𝐻(𝑤,𝑣,𝑛)   𝑉(𝑤)

Proof of Theorem numclwwlkovhOLD
StepHypRef Expression
1 oveq1 6800 . . . 4 (𝑛 = 𝑁 → (𝑛 ClWWalksN 𝐺) = (𝑁 ClWWalksN 𝐺))
21adantl 467 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → (𝑛 ClWWalksN 𝐺) = (𝑁 ClWWalksN 𝐺))
3 eqeq2 2782 . . . 4 (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
4 oveq1 6800 . . . . . 6 (𝑛 = 𝑁 → (𝑛 − 2) = (𝑁 − 2))
54fveq2d 6336 . . . . 5 (𝑛 = 𝑁 → (𝑤‘(𝑛 − 2)) = (𝑤‘(𝑁 − 2)))
65neeq1d 3002 . . . 4 (𝑛 = 𝑁 → ((𝑤‘(𝑛 − 2)) ≠ (𝑤‘0) ↔ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0)))
73, 6bi2anan9 620 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → (((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0)) ↔ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))))
82, 7rabeqbidv 3345 . 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 numclwwlkOLD.h . 2 𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})
10 ovex 6823 . . 3 (𝑁 ClWWalksN 𝐺) ∈ V
1110rabex 4946 . 2 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))} ∈ V
128, 9, 11ovmpt2a 6938 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 382   = wceq 1631  wcel 2145  wne 2943  {crab 3065  cfv 6031  (class class class)co 6793  cmpt2 6795  0cc0 10138  cmin 10468  cn 11222  2c2 11272  lastSclsw 13488  Vtxcvtx 26095   WWalksN cwwlksn 26954   ClWWalksN cclwwlkn 27174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798
This theorem is referenced by:  numclwwlk2lem1OLD  27574  numclwlk2lem2fOLD  27575  numclwlk2lem2f1oOLD  27577  numclwwlk3lemOLD  27580
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