Theorem iswspthsnon 29819

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, 12-May-2021.) (Revised by AV, 14-Mar-2022.)

Hypothesis

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

Assertion

Ref	Expression
iswspthsnon	⊢ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}

Distinct variable groups:   𝐴,𝑓,𝑤   𝐵,𝑓,𝑤   𝑓,𝐺,𝑤   𝑓,𝑁,𝑤   𝑓,𝑉,𝑤

Proof of Theorem iswspthsnon

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

Step	Hyp	Ref	Expression
1		0ov 7390	. . 3 ⊢ (𝐴∅𝐵) = ∅
2		df-wspthsnon 29797	. . . . 5 ⊢ 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	2	mpondm0 7593	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsNOn 𝐺) = ∅)
4	3	oveqd 7370	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = (𝐴∅𝐵))
5		id 22	. . . . . . 7 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → ¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V))
6	5	intnanrd 489	. . . . . 6 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → ¬ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
7		iswspthsnon.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
8	7	wwlksnon0 29817	. . . . . 6 ⊢ (¬ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
9	6, 8	syl 17	. . . . 5 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
10	9	rabeqdv 3412	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
11		rab0 4339	. . . 4 ⊢ {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = ∅
12	10, 11	eqtrdi 2780	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = ∅)
13	1, 4, 12	3eqtr4a 2790	. 2 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
14	7	wspthsnon 29815	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
15	14	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ ¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑁 WSPathsNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
16	15	oveqd 7370	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ ¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})𝐵))
17		eqid 2729	. . . . . . . 8 ⊢ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {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})
18	17	mpondm0 7593	. . . . . . 7 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})𝐵) = ∅)
19	18	adantl 481	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ ¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})𝐵) = ∅)
20	16, 19	eqtrd 2764	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ ¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅)
21	20	ex 412	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅))
22	4, 1	eqtrdi 2780	. . . . 5 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅)
23	22	a1d 25	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅))
24	21, 23	pm2.61i 182	. . 3 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅)
25	7	wwlksonvtx 29818	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
26	25	pm2.24d 151	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) → (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ¬ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
27	26	impcom 407	. . . . . 6 ⊢ ((¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵)) → ¬ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
28	27	nexdv 1936	. . . . 5 ⊢ ((¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵)) → ¬ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
29	28	ralrimiva 3121	. . . 4 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ∀𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ¬ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
30		rabeq0 4341	. . . 4 ⊢ ({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)
31	29, 30	sylibr 234	. . 3 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = ∅)
32	24, 31	eqtr4d 2767	. 2 ⊢ (¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
33	14	adantr 480	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑁 WSPathsNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
34		oveq12 7362	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎(𝑁 WWalksNOn 𝐺)𝑏) = (𝐴(𝑁 WWalksNOn 𝐺)𝐵))
35		oveq12 7362	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎(SPathsOn‘𝐺)𝑏) = (𝐴(SPathsOn‘𝐺)𝐵))
36	35	breqd 5106	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤 ↔ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
37	36	exbidv 1921	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤 ↔ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
38	34, 37	rabeqbidv 3415	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {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})
39	38	adantl 481	. . 3 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → {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})
40		simprl 770	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
41		simprr 772	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
42		ovex 7386	. . . . 5 ⊢ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∈ V
43	42	rabex 5281	. . . 4 ⊢ {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} ∈ V
44	43	a1i 11	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} ∈ V)
45	33, 39, 40, 41, 44	ovmpod 7505	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
46	13, 32, 45	ecase 1033	1 ⊢ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  {crab 3396  Vcvv 3438  ∅c0 4286   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  ℕ₀cn0 12402  Vtxcvtx 28959  SPathsOncspthson 29676   WWalksNOn cwwlksnon 29790   WSPathsNOn cwwspthsnon 29792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-wwlksnon 29795  df-wspthsnon 29797

This theorem is referenced by:  wspthnon  29821  wpthswwlks2on  29924