Theorem clwwlknonmpo 16874

Description: (ClWWalksNOn‘𝐺) is an operator mapping a vertex 𝑣 and a nonnegative integer 𝑛 to the set of closed walks on 𝑣 of length 𝑛 as words over the set of vertices in a graph 𝐺. (Contributed by AV, 25-Feb-2022.) (Proof shortened by AV, 2-Mar-2024.)

Assertion

Ref	Expression
clwwlknonmpo	⊢ (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})

Distinct variable group:   𝑛,𝐺,𝑣,𝑤

Proof of Theorem clwwlknonmpo

Dummy variables 𝑔 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clwwlknon 16873	. . . 4 ⊢ ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))
2	1	mptrcl 5791	. . 3 ⊢ (𝑥 ∈ (ClWWalksNOn‘𝐺) → 𝐺 ∈ V)
3		eqid 2238	. . . . 5 ⊢ (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
4	3	elmpom 6478	. . . 4 ⊢ (𝑥 ∈ (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) → ∃𝑠 𝑠 ∈ (Vtx‘𝐺))
5		df-vtx 16460	. . . . . 6 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
6	5	mptrcl 5791	. . . . 5 ⊢ (𝑠 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
7	6	exlimiv 1651	. . . 4 ⊢ (∃𝑠 𝑠 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
8	4, 7	syl 14	. . 3 ⊢ (𝑥 ∈ (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) → 𝐺 ∈ V)
9		fveq2 5698	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
10		eqidd 2239	. . . . . 6 ⊢ (𝑔 = 𝐺 → ℕ0 = ℕ0)
11		oveq2 6096	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑛 ClWWalksN 𝑔) = (𝑛 ClWWalksN 𝐺))
12	11	rabeqdv 2815	. . . . . 6 ⊢ (𝑔 = 𝐺 → {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
13	9, 10, 12	mpoeq123dv 6153	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
14		id 19	. . . . 5 ⊢ (𝐺 ∈ V → 𝐺 ∈ V)
15		vtxex 16464	. . . . . 6 ⊢ (𝐺 ∈ V → (Vtx‘𝐺) ∈ V)
16		nn0ex 9580	. . . . . 6 ⊢ ℕ0 ∈ V
17		mpoexga 6452	. . . . . 6 ⊢ (((Vtx‘𝐺) ∈ V ∧ ℕ0 ∈ V) → (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ∈ V)
18	15, 16, 17	sylancl 417	. . . . 5 ⊢ (𝐺 ∈ V → (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ∈ V)
19	1, 13, 14, 18	fvmptd3 5802	. . . 4 ⊢ (𝐺 ∈ V → (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
20	19	eleq2d 2308	. . 3 ⊢ (𝐺 ∈ V → (𝑥 ∈ (ClWWalksNOn‘𝐺) ↔ 𝑥 ∈ (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})))
21	2, 8, 20	pm5.21nii 716	. 2 ⊢ (𝑥 ∈ (ClWWalksNOn‘𝐺) ↔ 𝑥 ∈ (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
22	21	eqriv 2235	1 ⊢ (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})

This proof depends on syntax axioms:   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {crab 2532  Vcvv 2821  ifcif 3638   × cxp 4774  ‘cfv 5380  (class class class)co 6088   ∈ cmpo 6090  1^st c1st 6376  0cc0 8185  ℕ₀cn0 9574  Basecbs 13410  Vtxcvtx 16458   ClWWalksN cclwwlkn 16849  ClWWalksNOncclwwlknon 16872

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-i2m1 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-inn 9314  df-n0 9575  df-ndx 13413  df-slot 13414  df-base 13416  df-vtx 16460  df-clwwlknon 16873

This theorem is used by:  clwwlknon  16875  clwwlk0on0  16877