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Theorem av-numclwwlkovgel 41516
Description: Properties of an element of the value of operation 𝐶. (Contributed by Alexander van der Vekens, 24-Sep-2018.) (Revised by AV, 29-May-2021.)
Hypothesis
Ref Expression
av-numclwwlkovg.c 𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
Assertion
Ref Expression
av-numclwwlkovgel ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐶𝑁) ↔ (𝑃 ∈ (𝑁 ClWWalkSN 𝐺) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤   𝑤,𝑃
Allowed substitution hints:   𝐶(𝑤,𝑣,𝑛)   𝑃(𝑣,𝑛)   𝑉(𝑤)

Proof of Theorem av-numclwwlkovgel
StepHypRef Expression
1 av-numclwwlkovg.c . . . . 5 𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
21av-numclwwlkovg 41515 . . . 4 ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))})
32eleq2d 2667 . . 3 ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐶𝑁) ↔ 𝑃 ∈ {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))}))
4 fveq1 6082 . . . . . 6 (𝑤 = 𝑃 → (𝑤‘0) = (𝑃‘0))
54eqeq1d 2606 . . . . 5 (𝑤 = 𝑃 → ((𝑤‘0) = 𝑋 ↔ (𝑃‘0) = 𝑋))
6 fveq1 6082 . . . . . 6 (𝑤 = 𝑃 → (𝑤‘(𝑁 − 2)) = (𝑃‘(𝑁 − 2)))
76, 4eqeq12d 2619 . . . . 5 (𝑤 = 𝑃 → ((𝑤‘(𝑁 − 2)) = (𝑤‘0) ↔ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))
85, 7anbi12d 742 . . . 4 (𝑤 = 𝑃 → (((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0)) ↔ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))
98elrab 3325 . . 3 (𝑃 ∈ {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))} ↔ (𝑃 ∈ (𝑁 ClWWalkSN 𝐺) ∧ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))
103, 9syl6bb 274 . 2 ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐶𝑁) ↔ (𝑃 ∈ (𝑁 ClWWalkSN 𝐺) ∧ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))))
11 3anass 1034 . 2 ((𝑃 ∈ (𝑁 ClWWalkSN 𝐺) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)) ↔ (𝑃 ∈ (𝑁 ClWWalkSN 𝐺) ∧ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))
1210, 11syl6bbr 276 1 ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐶𝑁) ↔ (𝑃 ∈ (𝑁 ClWWalkSN 𝐺) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  {crab 2894  cfv 5785  (class class class)co 6522  cmpt2 6524  0cc0 9787  cmin 10112  2c2 10912  cuz 11514   ClWWalkSN cclwwlksn 41181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-iota 5749  df-fun 5787  df-fv 5793  df-ov 6525  df-oprab 6526  df-mpt2 6527
This theorem is referenced by:  av-numclwlk1lem2f1  41521
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