Theorem av-numclwwlkovgel 41519

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 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})

Assertion

Ref	Expression
av-numclwwlkovgel	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (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

Step	Hyp	Ref	Expression
1		av-numclwwlkovg.c	. . . . 5 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
2	1	av-numclwwlkovg 41518	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))})
3	2	eleq2d 2673	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑃 ∈ (𝑋𝐶𝑁) ↔ 𝑃 ∈ {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))}))
4		fveq1 6102	. . . . . 6 ⊢ (𝑤 = 𝑃 → (𝑤‘0) = (𝑃‘0))
5	4	eqeq1d 2612	. . . . 5 ⊢ (𝑤 = 𝑃 → ((𝑤‘0) = 𝑋 ↔ (𝑃‘0) = 𝑋))
6		fveq1 6102	. . . . . 6 ⊢ (𝑤 = 𝑃 → (𝑤‘(𝑁 − 2)) = (𝑃‘(𝑁 − 2)))
7	6, 4	eqeq12d 2625	. . . . 5 ⊢ (𝑤 = 𝑃 → ((𝑤‘(𝑁 − 2)) = (𝑤‘0) ↔ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))
8	5, 7	anbi12d 743	. . . 4 ⊢ (𝑤 = 𝑃 → (((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0)) ↔ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))
9	8	elrab 3331	. . 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))))
10	3, 9	syl6bb 275	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑃 ∈ (𝑋𝐶𝑁) ↔ (𝑃 ∈ (𝑁 ClWWalkSN 𝐺) ∧ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))))
11		3anass 1035	. 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))))
12	10, 11	syl6bbr 277	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑃 ∈ (𝑋𝐶𝑁) ↔ (𝑃 ∈ (𝑁 ClWWalkSN 𝐺) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815   − cmin 10145  2c2 10947  ℤ_≥cuz 11563   ClWWalkSN cclwwlksn 41184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  av-numclwlk1lem2f1  41524