Theorem clwwlknonmpo 16352

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 16351	. . . 4 ⊢ ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))
2	1	mptrcl 5738	. . 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 6412	. . . 4 ⊢ (𝑥 ∈ (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) → ∃𝑠 𝑠 ∈ (Vtx‘𝐺))
5		df-vtx 15938	. . . . . 6 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
6	5	mptrcl 5738	. . . . 5 ⊢ (𝑠 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
7	6	exlimiv 1647	. . . 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 5648	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
10		eqidd 2232	. . . . . 6 ⊢ (𝑔 = 𝐺 → ℕ0 = ℕ0)
11		oveq2 6036	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑛 ClWWalksN 𝑔) = (𝑛 ClWWalksN 𝐺))
12	11	rabeqdv 2797	. . . . . 6 ⊢ (𝑔 = 𝐺 → {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
13	9, 10, 12	mpoeq123dv 6093	. . . . 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 15942	. . . . . 6 ⊢ (𝐺 ∈ V → (Vtx‘𝐺) ∈ V)
16		nn0ex 9450	. . . . . 6 ⊢ ℕ0 ∈ V
17		mpoexga 6386	. . . . . 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 5749	. . . 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 712	. 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 1398  ∃wex 1541   ∈ wcel 2202  {crab 2515  Vcvv 2803  ifcif 3607   × cxp 4729  ‘cfv 5333  (class class class)co 6028   ∈ cmpo 6030  1^st c1st 6310  0cc0 8075  ℕ₀cn0 9444  Basecbs 13145  Vtxcvtx 15936   ClWWalksN cclwwlkn 16327  ClWWalksNOncclwwlknon 16350

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-i2m1 8180

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-inn 9186  df-n0 9445  df-ndx 13148  df-slot 13149  df-base 13151  df-vtx 15938  df-clwwlknon 16351

This theorem is referenced by:  clwwlknon  16353  clwwlk0on0  16355