Theorem wspthsn 29785

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 7399	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑛 WWalksN 𝑔) = (𝑁 WWalksN 𝐺))
2		fveq2 6861	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (SPaths‘𝑔) = (SPaths‘𝐺))
3	2	breqd 5121	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑓(SPaths‘𝑔)𝑤 ↔ 𝑓(SPaths‘𝐺)𝑤))
4	3	exbidv 1921	. . . . 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 3427	. . 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 29770	. . 3 ⊢ WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
8		ovex 7423	. . . 4 ⊢ (𝑁 WWalksN 𝐺) ∈ V
9	8	rabex 5297	. . 3 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} ∈ V
10	6, 7, 9	ovmpoa 7547	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
11	7	mpondm0 7632	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = ∅)
12		df-wwlksn 29768	. . . . . 6 ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
13	12	mpondm0 7632	. . . . 5 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
14	13	rabeqdv 3424	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
15		rab0 4352	. . . 4 ⊢ {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅
16	14, 15	eqtrdi 2781	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅)
17	11, 16	eqtr4d 2768	. 2 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
18	10, 17	pm2.61i 182	1 ⊢ (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤}

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {crab 3408  Vcvv 3450  ∅c0 4299   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  1c1 11076   + caddc 11078  ℕ₀cn0 12449  ♯chash 14302  SPathscspths 29648  WWalkscwwlks 29762   WWalksN cwwlksn 29763   WSPathsN cwwspthsn 29765

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-wwlksn 29768  df-wspthsn 29770

This theorem is referenced by:  iswspthn  29786  wspn0  29861