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

Theorem iswspthsnon 26982
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 6846 . . 3 (𝐴𝐵) = ∅
2 df-wspthsnon 26958 . . . . 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 7041 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsNOn 𝐺) = ∅)
43oveqd 6831 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = (𝐴𝐵))
5 id 22 . . . . . . 7 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → ¬ (𝑁 ∈ ℕ0𝐺 ∈ V))
65intnanrd 1001 . . . . . 6 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → ¬ ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))
7 iswspthsnon.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
87wwlksnon0 26980 . . . . . 6 (¬ ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
96, 8syl 17 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
109rabeqdv 3334 . . . 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 4098 . . . 4 {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = ∅
1210, 11syl6eq 2810 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = ∅)
131, 4, 123eqtr4a 2820 . 2 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
147wspthsnon 26977 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
1514adantr 472 . . . . . . 7 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
1615oveqd 6831 . . . . . 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 2760 . . . . . . . 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 7041 . . . . . . 7 (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})𝐵) = ∅)
1918adantl 473 . . . . . 6 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})𝐵) = ∅)
2016, 19eqtrd 2794 . . . . 5 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ ¬ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅)
2120ex 449 . . . 4 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = ∅))
224, 1syl6eq 2810 . . . . 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 26981 . . . . . . . . 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 2004 . . . . . 6 ((¬ (𝐴𝑉𝐵𝑉) ∧ 𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵)) → ∀𝑓 ¬ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
29 alnex 1855 . . . . . 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 3104 . . . 4 (¬ (𝐴𝑉𝐵𝑉) → ∀𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ¬ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
32 rabeq0 4100 . . . 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 2797 . 2 (¬ (𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
3514adantr 472 . . 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 6823 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎(𝑁 WWalksNOn 𝐺)𝑏) = (𝐴(𝑁 WWalksNOn 𝐺)𝐵))
37 oveq12 6823 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎(SPathsOn‘𝐺)𝑏) = (𝐴(SPathsOn‘𝐺)𝐵))
3837breqd 4815 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
3938exbidv 1999 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤 ↔ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
4036, 39rabeqbidv 3335 . . . 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 473 . . 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 811 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → 𝐴𝑉)
43 simprr 813 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → 𝐵𝑉)
44 ovex 6842 . . . . 5 (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∈ V
4544rabex 4964 . . . 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 6954 . 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 1020 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 1630   = wceq 1632  wex 1853  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  c0 4058   class class class wbr 4804  cfv 6049  (class class class)co 6814  cmpt2 6816  0cn0 11504  Vtxcvtx 26094  SPathsOncspthson 26842   WWalksNOn cwwlksnon 26951   WSPathsNOn cwwspthsnon 26953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-wwlksnon 26956  df-wspthsnon 26958
This theorem is referenced by:  wspthnon  26985  wspthnonOLD  26987  wpthswwlks2on  27103
  Copyright terms: Public domain W3C validator