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Theorem iswspthsnonOLD 26983
Description: Obsolete version of iswspthsnon 26982 as of 14-Mar-2022. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 12-May-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
iswspthsnon.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
iswspthsnonOLD ((𝐴𝑉𝐵𝑉) → (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 iswspthsnonOLD
Dummy variables 𝑎 𝑏 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iswspthsnon.v . . . . . 6 𝑉 = (Vtx‘𝐺)
21wspthsnon 26977 . . . . 5 ((𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
32adantr 472 . . . 4 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
4 oveq12 6823 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎(𝑁 WWalksNOn 𝐺)𝑏) = (𝐴(𝑁 WWalksNOn 𝐺)𝐵))
5 oveq12 6823 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎(SPathsOn‘𝐺)𝑏) = (𝐴(SPathsOn‘𝐺)𝐵))
65breqd 4815 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
76exbidv 1999 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤 ↔ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
84, 7rabeqbidv 3335 . . . . 5 ((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})
98adantl 473 . . . 4 ((((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})
10 simprl 811 . . . 4 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → 𝐴𝑉)
11 simprr 813 . . . 4 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → 𝐵𝑉)
12 ovex 6842 . . . . . 6 (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∈ V
1312rabex 4964 . . . . 5 {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} ∈ V
1413a1i 11 . . . 4 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} ∈ V)
153, 9, 10, 11, 14ovmpt2d 6954 . . 3 (((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
1615ex 449 . 2 ((𝑁 ∈ ℕ0𝐺 ∈ V) → ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}))
17 0ov 6846 . . . 4 (𝐴𝐵) = ∅
18 df-wspthsnon 26958 . . . . . 6 WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
1918mpt2ndm0 7041 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WSPathsNOn 𝐺) = ∅)
2019oveqd 6831 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = (𝐴𝐵))
21 df-wwlksnon 26956 . . . . . . . . 9 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
2221mpt2ndm0 7041 . . . . . . . 8 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝑁 WWalksNOn 𝐺) = ∅)
2322oveqd 6831 . . . . . . 7 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴𝐵))
2423, 17syl6eq 2810 . . . . . 6 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
2524rabeqdv 3334 . . . . 5 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
26 rab0 4098 . . . . 5 {𝑤 ∈ ∅ ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = ∅
2725, 26syl6eq 2810 . . . 4 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = ∅)
2817, 20, 273eqtr4a 2820 . . 3 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
2928a1d 25 . 2 (¬ (𝑁 ∈ ℕ0𝐺 ∈ V) → ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}))
3016, 29pm2.61i 176 1 ((𝐴𝑉𝐵𝑉) → (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   = wceq 1632  wex 1853  wcel 2139  {crab 3054  Vcvv 3340  c0 4058   class class class wbr 4804  cfv 6049  (class class class)co 6814  cmpt2 6816  0cc0 10148  0cn0 11504  Vtxcvtx 26094  SPathsOncspthson 26842   WWalksN cwwlksn 26950   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-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:  wspthnonOLDOLD  26988  wpthswwlks2onOLD  27104
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