Theorem wspthsn 28114

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 7264	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑛 WWalksN 𝑔) = (𝑁 WWalksN 𝐺))
2		fveq2 6756	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (SPaths‘𝑔) = (SPaths‘𝐺))
3	2	breqd 5081	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑓(SPaths‘𝑔)𝑤 ↔ 𝑓(SPaths‘𝐺)𝑤))
4	3	exbidv 1925	. . . . 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 3410	. . 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 28099	. . 3 ⊢ WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
8		ovex 7288	. . . 4 ⊢ (𝑁 WWalksN 𝐺) ∈ V
9	8	rabex 5251	. . 3 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} ∈ V
10	6, 7, 9	ovmpoa 7406	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
11	7	mpondm0 7488	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = ∅)
12		df-wwlksn 28097	. . . . . 6 ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
13	12	mpondm0 7488	. . . . 5 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
14	13	rabeqdv 3409	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
15		rab0 4313	. . . 4 ⊢ {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅
16	14, 15	eqtrdi 2795	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅)
17	11, 16	eqtr4d 2781	. 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 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {crab 3067  Vcvv 3422  ∅c0 4253   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  1c1 10803   + caddc 10805  ℕ₀cn0 12163  ♯chash 13972  SPathscspths 27982  WWalkscwwlks 28091   WWalksN cwwlksn 28092   WSPathsN cwwspthsn 28094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-wwlksn 28097  df-wspthsn 28099

This theorem is referenced by:  iswspthn  28115  wspn0  28190