Theorem wspthsnon 30498

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 30480	. . 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 6916	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4		wwlksnon.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
5	3, 4	eqtr4di 2816	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
6	5	adantl 487	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (Vtx‘𝑔) = 𝑉)
7		oveq12 7460	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑛 WWalksNOn 𝑔) = (𝑁 WWalksNOn 𝐺))
8	7	oveqd 7468	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑎(𝑛 WWalksNOn 𝑔)𝑏) = (𝑎(𝑁 WWalksNOn 𝐺)𝑏))
9		fveq2 6916	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (SPathsOn‘𝑔) = (SPathsOn‘𝐺))
10	9	oveqd 7468	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑎(SPathsOn‘𝑔)𝑏) = (𝑎(SPathsOn‘𝐺)𝑏))
11	10	breqd 5119	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤 ↔ 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
12	11	adantl 487	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤 ↔ 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
13	12	exbidv 1956	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤 ↔ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
14	8, 13	rabeqbidv 3433	. . . 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 7526	. . 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 487	. 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 488	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → 𝑁 ∈ ℕ0)
18		elex 3475	. . 3 ⊢ (𝐺 ∈ 𝑈 → 𝐺 ∈ V)
19	18	adantl 487	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → 𝐺 ∈ V)
20	4	fvexi 6930	. . . 4 ⊢ 𝑉 ∈ V
21	20, 20	mpoex 8127	. . 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 7604	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}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {crab 3416  Vcvv 3454   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  ℕ₀cn0 12646  Vtxcvtx 29631  SPathsOncspthson 30355   WWalksNOn cwwlksnon 30473   WSPathsNOn cwwspthsnon 30475

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-wspthsnon 30480

This theorem is used by:  iswspthsnon  30502