Theorem wspthsnon 27624

Description: The set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)

Hypothesis

Ref	Expression
wwlksnon.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
wspthsnon	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))

Distinct variable groups:   𝐺,𝑎,𝑏,𝑤   𝑁,𝑎,𝑏,𝑤   𝑉,𝑎,𝑏   𝑓,𝐺,𝑎,𝑏,𝑤   𝑓,𝑁

Allowed substitution hints:   𝑈(𝑤,𝑓,𝑎,𝑏)   𝑉(𝑤,𝑓)

Proof of Theorem wspthsnon

Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wspthsnon 27606	. . 3 ⊢ WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
2	1	a1i 11	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤})))
3		fveq2 6665	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4		wwlksnon.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
5	3, 4	syl6eqr 2874	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
6	5	adantl 484	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (Vtx‘𝑔) = 𝑉)
7		oveq12 7159	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑛 WWalksNOn 𝑔) = (𝑁 WWalksNOn 𝐺))
8	7	oveqd 7167	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑎(𝑛 WWalksNOn 𝑔)𝑏) = (𝑎(𝑁 WWalksNOn 𝐺)𝑏))
9		fveq2 6665	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (SPathsOn‘𝑔) = (SPathsOn‘𝐺))
10	9	oveqd 7167	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑎(SPathsOn‘𝑔)𝑏) = (𝑎(SPathsOn‘𝐺)𝑏))
11	10	breqd 5070	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤 ↔ 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
12	11	adantl 484	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤 ↔ 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
13	12	exbidv 1918	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤 ↔ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
14	8, 13	rabeqbidv 3486	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤} = {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})
15	6, 6, 14	mpoeq123dv 7223	. . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
16	15	adantl 484	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ (𝑛 = 𝑁 ∧ 𝑔 = 𝐺)) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
17		simpl 485	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → 𝑁 ∈ ℕ0)
18		elex 3513	. . 3 ⊢ (𝐺 ∈ 𝑈 → 𝐺 ∈ V)
19	18	adantl 484	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → 𝐺 ∈ V)
20	4	fvexi 6679	. . . 4 ⊢ 𝑉 ∈ V
21	20, 20	mpoex 7771	. . 3 ⊢ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}) ∈ V
22	21	a1i 11	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}) ∈ V)
23	2, 16, 17, 19, 22	ovmpod 7296	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {crab 3142  Vcvv 3495   class class class wbr 5059  ‘cfv 6350  (class class class)co 7150   ∈ cmpo 7152  ℕ₀cn0 11891  Vtxcvtx 26775  SPathsOncspthson 27490   WWalksNOn cwwlksnon 27599   WSPathsNOn cwwspthsnon 27601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-wspthsnon 27606

This theorem is referenced by:  iswspthsnon  27628