MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iswspthsnon Structured version   Visualization version   GIF version

Theorem iswspthsnon 26806
Description: The set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
Hypothesis
Ref Expression
iswspthsnon.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
iswspthsnon (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}
Distinct variable groups:   𝐴,𝑓,𝑤   𝐵,𝑓,𝑤   𝑓,𝐺,𝑤   𝑓,𝑁,𝑤   𝑓,𝑉,𝑤

Proof of Theorem iswspthsnon
Dummy variables 𝑎 𝑏 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ov 6722 . . 3 (𝐴𝐵) = ∅
2 df-wspthsnon 26782 . . . . 5 WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
32mpt2ndm0 6917 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsNOn 𝐺) = ∅)
43oveqd 6707 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = (𝐴𝐵))
5 id 22 . . . . . . 7 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → ¬ (𝑁 ∈ ℕ0𝐺 ∈ V))
65intnanrd 983 . . . . . 6 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → ¬ ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))
7 iswspthsnon.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
87wwlksnon0 26804 . . . . . 6 (¬ ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
96, 8syl 17 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
109rabeqdv 3225 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
11 rab0 3988 . . . 4 {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = ∅
1210, 11syl6eq 2701 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = ∅)
131, 4, 123eqtr4a 2711 . 2 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
147wspthsnon 26801 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
1514adantr 480 . . . . . . 7 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
1615oveqd 6707 . . . . . 6 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})𝐵))
17 eqid 2651 . . . . . . . 8 (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})
1817mpt2ndm0 6917 . . . . . . 7 (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})𝐵) = ∅)
1918adantl 481 . . . . . 6 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})𝐵) = ∅)
2016, 19eqtrd 2685 . . . . 5 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅)
2120ex 449 . . . 4 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅))
224, 1syl6eq 2701 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅)
2322a1d 25 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅))
2421, 23pm2.61i 176 . . 3 (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅)
257wwlksonvtx 26805 . . . . . . . . 9 (𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) → (𝐴𝑉𝐵𝑉))
2625pm2.24d 147 . . . . . . . 8 (𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) → (¬ (𝐴𝑉𝐵𝑉) → ¬ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
2726impcom 445 . . . . . . 7 ((¬ (𝐴𝑉𝐵𝑉) ∧ 𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵)) → ¬ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
2827alrimiv 1895 . . . . . 6 ((¬ (𝐴𝑉𝐵𝑉) ∧ 𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵)) → ∀𝑓 ¬ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
29 alnex 1746 . . . . . 6 (∀𝑓 ¬ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤 ↔ ¬ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
3028, 29sylib 208 . . . . 5 ((¬ (𝐴𝑉𝐵𝑉) ∧ 𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵)) → ¬ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
3130ralrimiva 2995 . . . 4 (¬ (𝐴𝑉𝐵𝑉) → ∀𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ¬ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
32 rabeq0 3990 . . . 4 ({𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = ∅ ↔ ∀𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ¬ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
3331, 32sylibr 224 . . 3 (¬ (𝐴𝑉𝐵𝑉) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = ∅)
3424, 33eqtr4d 2688 . 2 (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
3514adantr 480 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
36 oveq12 6699 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎(𝑁 WWalksNOn 𝐺)𝑏) = (𝐴(𝑁 WWalksNOn 𝐺)𝐵))
37 oveq12 6699 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎(SPathsOn‘𝐺)𝑏) = (𝐴(SPathsOn‘𝐺)𝐵))
3837breqd 4696 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3938exbidv 1890 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤 ↔ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
4036, 39rabeqbidv 3226 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤} = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
4140adantl 481 . . 3 ((((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤} = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
42 simprl 809 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → 𝐴𝑉)
43 simprr 811 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → 𝐵𝑉)
44 ovex 6718 . . . . 5 (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∈ V
4544rabex 4845 . . . 4 {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} ∈ V
4645a1i 11 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} ∈ V)
4735, 41, 42, 43, 46ovmpt2d 6830 . 2 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
4813, 34, 47ecase 1002 1 (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  c0 3948   class class class wbr 4685  cfv 5926  (class class class)co 6690  cmpt2 6692  0cn0 11330  Vtxcvtx 25919  SPathsOncspthson 26667   WWalksNOn cwwlksnon 26775   WSPathsNOn cwwspthsnon 26777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-wwlksnon 26780  df-wspthsnon 26782
This theorem is referenced by:  wspthnon  26809  wspthnonOLD  26811  wpthswwlks2on  26927
  Copyright terms: Public domain W3C validator