Theorem 2clwwlkel 30951

Description: Characterization of an element of the value of operation 𝐶, i.e., of a word being a double loop of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 24-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 20-Apr-2022.)

Hypothesis

Ref	Expression
2clwwlk.c	⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})

Assertion

Ref	Expression
2clwwlkel	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑊 ∈ (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   𝑤,𝑊

Allowed substitution hints:   𝐶(𝑤, 𝑣, 𝑛)   𝑉(𝑤)   𝑊(𝑣, 𝑛)

Proof of Theorem 2clwwlkel

Step	Hyp	Ref	Expression
1		2clwwlk.c	. . . 4 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
2	1	2clwwlk 30949	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})
3	2	eleq2d 2846	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑊 ∈ (𝑋𝐶𝑁) ↔ 𝑊 ∈ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}))
4		fveq1 6884	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤‘(𝑁 − 2)) = (𝑊‘(𝑁 − 2)))
5	4	eqeq1d 2762	. . 3 ⊢ (𝑤 = 𝑊 → ((𝑤‘(𝑁 − 2)) = 𝑋 ↔ (𝑊‘(𝑁 − 2)) = 𝑋))
6	5	elrab 3644	. 2 ⊢ (𝑊 ∈ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ↔ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋))
7	3, 6	bitrdi 290	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑊 ∈ (𝑋𝐶𝑁) ↔ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {crab 3412  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422   − cmin 11542  2c2 12398  ℤ_≥cuz 12966  ClWWalksNOncclwwlknon 30679

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is used by:  2clwwlk2clwwlk  30952  numclwwlk1lem2f1  30959