Theorem wspthsn 29921

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 7367	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑛 WWalksN 𝑔) = (𝑁 WWalksN 𝐺))
2		fveq2 6834	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (SPaths‘𝑔) = (SPaths‘𝐺))
3	2	breqd 5109	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑓(SPaths‘𝑔)𝑤 ↔ 𝑓(SPaths‘𝐺)𝑤))
4	3	exbidv 1922	. . . . 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 3417	. . 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 29906	. . 3 ⊢ WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
8		ovex 7391	. . . 4 ⊢ (𝑁 WWalksN 𝐺) ∈ V
9	8	rabex 5284	. . 3 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} ∈ V
10	6, 7, 9	ovmpoa 7513	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
11	7	mpondm0 7598	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = ∅)
12		df-wwlksn 29904	. . . . . 6 ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
13	12	mpondm0 7598	. . . . 5 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
14	13	rabeqdv 3414	. . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
15		rab0 4338	. . . 4 ⊢ {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅
16	14, 15	eqtrdi 2787	. . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅)
17	11, 16	eqtr4d 2774	. 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 1541  ∃wex 1780   ∈ wcel 2113  {crab 3399  Vcvv 3440  ∅c0 4285   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  1c1 11027   + caddc 11029  ℕ₀cn0 12401  ♯chash 14253  SPathscspths 29784  WWalkscwwlks 29898   WWalksN cwwlksn 29899   WSPathsN cwwspthsn 29901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-wwlksn 29904  df-wspthsn 29906

This theorem is referenced by:  iswspthn  29922  wspn0  29997