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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 6803 . . . 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 6803 . . . . . 6 (𝑛 = 𝑁 → (𝑛 − 2) = (𝑁 − 2))
54fveq2d 6337 . . . . 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 6827 . . 3 (𝑁 ClWWalksN 𝐺) ∈ V
1110rabex 4947 . 2 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))} ∈ V
128, 9, 11ovmpt2a 6942 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 6030  (class class class)co 6796  cmpt2 6798  0cc0 10142  cmin 10472  cn 11226  2c2 11276  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 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801
This theorem is referenced by:  numclwwlk2lem1OLD  27574  numclwlk2lem2fOLD  27575  numclwlk2lem2f1oOLD  27577  numclwwlk3lemOLD  27580
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