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Theorem wspthsn 26621
 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.
StepHypRef Expression
1 oveq12 6619 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑛 WWalksN 𝑔) = (𝑁 WWalksN 𝐺))
2 fveq2 6153 . . . . . . 7 (𝑔 = 𝐺 → (SPaths‘𝑔) = (SPaths‘𝐺))
32breqd 4629 . . . . . 6 (𝑔 = 𝐺 → (𝑓(SPaths‘𝑔)𝑤𝑓(SPaths‘𝐺)𝑤))
43exbidv 1847 . . . . 5 (𝑔 = 𝐺 → (∃𝑓 𝑓(SPaths‘𝑔)𝑤 ↔ ∃𝑓 𝑓(SPaths‘𝐺)𝑤))
54adantl 482 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → (∃𝑓 𝑓(SPaths‘𝑔)𝑤 ↔ ∃𝑓 𝑓(SPaths‘𝐺)𝑤))
61, 5rabeqbidv 3184 . . 3 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
7 df-wspthsn 26611 . . 3 WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
8 ovex 6638 . . . 4 (𝑁 WWalksN 𝐺) ∈ V
98rabex 4778 . . 3 {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} ∈ V
106, 7, 9ovmpt2a 6751 . 2 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
117mpt2ndm0 6835 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = ∅)
12 df-wwlksn 26609 . . . . . 6 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)})
1312mpt2ndm0 6835 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = ∅)
1413rabeqdv 3183 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
15 rab0 3934 . . . 4 {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅
1614, 15syl6eq 2671 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅)
1711, 16eqtr4d 2658 . 2 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤})
1810, 17pm2.61i 176 1 (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987  {crab 2911  Vcvv 3189  ∅c0 3896   class class class wbr 4618  ‘cfv 5852  (class class class)co 6610  1c1 9889   + caddc 9891  ℕ0cn0 11244  #chash 13065  SPathscspths 26495  WWalkscwwlks 26603   WWalksN cwwlksn 26604   WSPathsN cwwspthsn 26606 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-wwlksn 26609  df-wspthsn 26611 This theorem is referenced by:  iswspthn  26622  wspn0  26706
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