Theorem wspthsn 41046

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

Assertion

Ref	Expression
wspthsn	⊢ (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤}

Distinct variable groups:   𝑓,𝐺,𝑤   𝑤,𝑁

Allowed substitution hint:   𝑁(𝑓)

Proof of Theorem wspthsn

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

Step	Hyp	Ref	Expression
1		oveq12 6558	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑛 WWalkSN 𝑔) = (𝑁 WWalkSN 𝐺))
2		fveq2 6103	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (SPathS‘𝑔) = (SPathS‘𝐺))
3	2	breqd 4594	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑓(SPathS‘𝑔)𝑤 ↔ 𝑓(SPathS‘𝐺)𝑤))
4	3	exbidv 1837	. . . . 5 ⊢ (𝑔 = 𝐺 → (∃𝑓 𝑓(SPathS‘𝑔)𝑤 ↔ ∃𝑓 𝑓(SPathS‘𝐺)𝑤))
5	4	adantl 481	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (∃𝑓 𝑓(SPathS‘𝑔)𝑤 ↔ ∃𝑓 𝑓(SPathS‘𝐺)𝑤))
6	1, 5	rabeqbidv 3168	. . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ∃𝑓 𝑓(SPathS‘𝑔)𝑤} = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤})
7		df-wspthsn 41036	. . 3 ⊢ WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ∃𝑓 𝑓(SPathS‘𝑔)𝑤})
8		ovex 6577	. . . 4 ⊢ (𝑁 WWalkSN 𝐺) ∈ V
9	8	rabex 4740	. . 3 ⊢ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤} ∈ V
10	6, 7, 9	ovmpt2a 6689	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤})
11	7	mpt2ndm0 6773	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = ∅)
12		df-wwlksn 41034	. . . . . 6 ⊢ WWalkSN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalkS‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)})
13	12	mpt2ndm0 6773	. . . . 5 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalkSN 𝐺) = ∅)
14	13	rabeqdv 3167	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤} = {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤})
15		rab0 3909	. . . 4 ⊢ {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤} = ∅
16	14, 15	syl6eq 2660	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤} = ∅)
17	11, 16	eqtr4d 2647	. 2 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤})
18	10, 17	pm2.61i 175	1 ⊢ (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤}

Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900  Vcvv 3173  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818  ℕ₀cn0 11169  #chash 12979  SPathScspths 40920  WWalkScwwlks 41028   WWalkSN cwwlksn 41029   WSPathsN cwwspthsn 41031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wwlksn 41034  df-wspthsn 41036

This theorem is referenced by:  iswspthn  41047  wspn0  41131