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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  ℕ0cn0 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
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