Theorem numclwwlk3lem2lem 30319

Description: Lemma for numclwwlk3lem2 30320: The set of closed vertices of a fixed length 𝑁 on a fixed vertex 𝑉 is the union of the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex being 𝑉 and the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex not being 𝑉. (Contributed by AV, 1-May-2022.)

Hypotheses

Ref	Expression
numclwwlk3lem2.c	⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
numclwwlk3lem2.h	⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})

Assertion

Ref	Expression
numclwwlk3lem2lem	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ((𝑋𝐻𝑁) ∪ (𝑋𝐶𝑁)))

Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤

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

Proof of Theorem numclwwlk3lem2lem

Step	Hyp	Ref	Expression
1		numclwwlk3lem2.h	. . . 4 ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
2	1	numclwwlkovh0 30308	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋})
3		numclwwlk3lem2.c	. . . 4 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
4	3	2clwwlk 30283	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})
5	2, 4	uneq12d 4140	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → ((𝑋𝐻𝑁) ∪ (𝑋𝐶𝑁)) = ({𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋} ∪ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}))
6		unrab 4286	. . 3 ⊢ ({𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋} ∪ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ ((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋)}
7		exmidne 2937	. . . . . 6 ⊢ ((𝑤‘(𝑁 − 2)) = 𝑋 ∨ (𝑤‘(𝑁 − 2)) ≠ 𝑋)
8		orcom 870	. . . . . 6 ⊢ (((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋) ↔ ((𝑤‘(𝑁 − 2)) = 𝑋 ∨ (𝑤‘(𝑁 − 2)) ≠ 𝑋))
9	7, 8	mpbir 231	. . . . 5 ⊢ ((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋)
10	9	a1i 11	. . . 4 ⊢ (𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) → ((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋))
11	10	rabeqc 3424	. . 3 ⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ ((𝑤‘(𝑁 − 2)) ≠ 𝑋 ∨ (𝑤‘(𝑁 − 2)) = 𝑋)} = (𝑋(ClWWalksNOn‘𝐺)𝑁)
12	6, 11	eqtri 2753	. 2 ⊢ ({𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋} ∪ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) = (𝑋(ClWWalksNOn‘𝐺)𝑁)
13	5, 12	eqtr2di 2782	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ((𝑋𝐻𝑁) ∪ (𝑋𝐶𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2927  {crab 3411   ∪ cun 3920  ‘cfv 6519  (class class class)co 7394   ∈ cmpo 7396   − cmin 11423  2c2 12252  ℤ_≥cuz 12809  ClWWalksNOncclwwlknon 30023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-sbc 3762  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-iota 6472  df-fun 6521  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399

This theorem is referenced by:  numclwwlk3lem2  30320