Theorem clwwlknonmpo 16278

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 16277	. . . 4 ⊢ ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))
2	1	mptrcl 5729	. . 3 ⊢ (𝑥 ∈ (ClWWalksNOn‘𝐺) → 𝐺 ∈ V)
3		eqid 2231	. . . . 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 6402	. . . 4 ⊢ (𝑥 ∈ (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) → ∃𝑠 𝑠 ∈ (Vtx‘𝐺))
5		df-vtx 15864	. . . . . 6 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
6	5	mptrcl 5729	. . . . 5 ⊢ (𝑠 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
7	6	exlimiv 1646	. . . 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 5639	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
10		eqidd 2232	. . . . . 6 ⊢ (𝑔 = 𝐺 → ℕ0 = ℕ0)
11		oveq2 6025	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑛 ClWWalksN 𝑔) = (𝑛 ClWWalksN 𝐺))
12	11	rabeqdv 2796	. . . . . 6 ⊢ (𝑔 = 𝐺 → {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
13	9, 10, 12	mpoeq123dv 6082	. . . . 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 15868	. . . . . 6 ⊢ (𝐺 ∈ V → (Vtx‘𝐺) ∈ V)
16		nn0ex 9407	. . . . . 6 ⊢ ℕ0 ∈ V
17		mpoexga 6376	. . . . . 6 ⊢ (((Vtx‘𝐺) ∈ V ∧ ℕ0 ∈ V) → (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ∈ V)
18	15, 16, 17	sylancl 413	. . . . 5 ⊢ (𝐺 ∈ V → (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ∈ V)
19	1, 13, 14, 18	fvmptd3 5740	. . . 4 ⊢ (𝐺 ∈ V → (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
20	19	eleq2d 2301	. . 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 711	. 2 ⊢ (𝑥 ∈ (ClWWalksNOn‘𝐺) ↔ 𝑥 ∈ (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
22	21	eqriv 2228	1 ⊢ (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})

Syntax hints:   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {crab 2514  Vcvv 2802  ifcif 3605   × cxp 4723  ‘cfv 5326  (class class class)co 6017   ∈ cmpo 6019  1^st c1st 6300  0cc0 8031  ℕ₀cn0 9401  Basecbs 13081  Vtxcvtx 15862   ClWWalksN cclwwlkn 16253  ClWWalksNOncclwwlknon 16276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-i2m1 8136

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-inn 9143  df-n0 9402  df-ndx 13084  df-slot 13085  df-base 13087  df-vtx 15864  df-clwwlknon 16277

This theorem is referenced by:  clwwlknon  16279  clwwlk0on0  16281