Theorem numclwwlkovh0 30459

Description: Value of operation 𝐻, mapping a vertex 𝑣 and an integer 𝑛 greater than 1 to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v" according to definition 7 in [Huneke] p. 2. (Contributed by AV, 1-May-2022.)

Hypothesis

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

Assertion

Ref	Expression
numclwwlkovh0	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐻𝑁) = {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

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

Proof of Theorem numclwwlkovh0

Step	Hyp	Ref	Expression
1		oveq12 7377	. . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (𝑣(ClWWalksNOn‘𝐺)𝑛) = (𝑋(ClWWalksNOn‘𝐺)𝑁))
2		oveq1 7375	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 − 2) = (𝑁 − 2))
3	2	adantl 481	. . . . 5 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (𝑛 − 2) = (𝑁 − 2))
4	3	fveq2d 6846	. . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (𝑤‘(𝑛 − 2)) = (𝑤‘(𝑁 − 2)))
5		simpl 482	. . . 4 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → 𝑣 = 𝑋)
6	4, 5	neeq12d 2994	. . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → ((𝑤‘(𝑛 − 2)) ≠ 𝑣 ↔ (𝑤‘(𝑁 − 2)) ≠ 𝑋))
7	1, 6	rabeqbidv 3419	. 2 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣} = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋})
8		numclwwlkovh.h	. 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
9		ovex 7401	. . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V
10	9	rabex 5286	. 2 ⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋} ∈ V
11	7, 8, 10	ovmpoa 7523	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3401  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370   − cmin 11376  2c2 12212  ℤ_≥cuz 12763  ClWWalksNOncclwwlknon 30174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373

This theorem is referenced by:  numclwwlkovh  30460  numclwwlk3lem2lem  30470  numclwwlk3lem2  30471