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Theorem wspthsnon 26956
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, 11-May-2021.)
Hypothesis
Ref Expression
wwlksnon.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
wspthsnon ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
Distinct variable groups:   𝐺,𝑎,𝑏,𝑤   𝑁,𝑎,𝑏,𝑤   𝑉,𝑎,𝑏   𝑓,𝐺,𝑎,𝑏,𝑤   𝑓,𝑁
Allowed substitution hints:   𝑈(𝑤,𝑓,𝑎,𝑏)   𝑉(𝑤,𝑓)

Proof of Theorem wspthsnon
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wspthsnon 26937 . . 3 WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
21a1i 11 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤})))
3 fveq2 6352 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4 wwlksnon.v . . . . . 6 𝑉 = (Vtx‘𝐺)
53, 4syl6eqr 2812 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
65adantl 473 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → (Vtx‘𝑔) = 𝑉)
7 oveq12 6822 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑛 WWalksNOn 𝑔) = (𝑁 WWalksNOn 𝐺))
87oveqd 6830 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑎(𝑛 WWalksNOn 𝑔)𝑏) = (𝑎(𝑁 WWalksNOn 𝐺)𝑏))
9 fveq2 6352 . . . . . . . . 9 (𝑔 = 𝐺 → (SPathsOn‘𝑔) = (SPathsOn‘𝐺))
109oveqd 6830 . . . . . . . 8 (𝑔 = 𝐺 → (𝑎(SPathsOn‘𝑔)𝑏) = (𝑎(SPathsOn‘𝐺)𝑏))
1110breqd 4815 . . . . . . 7 (𝑔 = 𝐺 → (𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
1211adantl 473 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
1312exbidv 1999 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → (∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤 ↔ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤))
148, 13rabeqbidv 3335 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → {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})
156, 6, 14mpt2eq123dv 6882 . . 3 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {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}))
1615adantl 473 . 2 (((𝑁 ∈ ℕ0𝐺𝑈) ∧ (𝑛 = 𝑁𝑔 = 𝐺)) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {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}))
17 simpl 474 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → 𝑁 ∈ ℕ0)
18 elex 3352 . . 3 (𝐺𝑈𝐺 ∈ V)
1918adantl 473 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → 𝐺 ∈ V)
20 fvex 6362 . . . . 5 (Vtx‘𝐺) ∈ V
214, 20eqeltri 2835 . . . 4 𝑉 ∈ V
2221, 21mpt2ex 7415 . . 3 (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}) ∈ V
2322a1i 11 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}) ∈ V)
242, 16, 17, 19, 23ovmpt2d 6953 1 ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  {crab 3054  Vcvv 3340   class class class wbr 4804  cfv 6049  (class class class)co 6813  cmpt2 6815  0cn0 11484  Vtxcvtx 26073  SPathsOncspthson 26821   WWalksNOn cwwlksnon 26930   WSPathsNOn cwwspthsnon 26932
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 7114
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 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-wspthsnon 26937
This theorem is referenced by:  iswspthsnon  26961  iswspthsnonOLD  26962
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