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Theorem is2wlkonot 26390
 Description: The set of walks of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
Assertion
Ref Expression
is2wlkonot ((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOnOt 𝐸) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))
Distinct variable groups:   𝐸,𝑎,𝑏,𝑡,𝑓,𝑝   𝑉,𝑎,𝑏,𝑡,𝑓,𝑝
Allowed substitution hints:   𝑋(𝑡,𝑓,𝑝,𝑎,𝑏)   𝑌(𝑡,𝑓,𝑝,𝑎,𝑏)

Proof of Theorem is2wlkonot
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3185 . . 3 (𝑉𝑋𝑉 ∈ V)
21adantr 480 . 2 ((𝑉𝑋𝐸𝑌) → 𝑉 ∈ V)
3 elex 3185 . . 3 (𝐸𝑌𝐸 ∈ V)
43adantl 481 . 2 ((𝑉𝑋𝐸𝑌) → 𝐸 ∈ V)
5 mpt2exga 7135 . . . 4 ((𝑉𝑋𝑉𝑋) → (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}) ∈ V)
65anidms 675 . . 3 (𝑉𝑋 → (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}) ∈ V)
76adantr 480 . 2 ((𝑉𝑋𝐸𝑌) → (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}) ∈ V)
8 simpl 472 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
9 id 22 . . . . . . . 8 (𝑣 = 𝑉𝑣 = 𝑉)
109, 9xpeq12d 5064 . . . . . . 7 (𝑣 = 𝑉 → (𝑣 × 𝑣) = (𝑉 × 𝑉))
1110, 9xpeq12d 5064 . . . . . 6 (𝑣 = 𝑉 → ((𝑣 × 𝑣) × 𝑣) = ((𝑉 × 𝑉) × 𝑉))
1211adantr 480 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → ((𝑣 × 𝑣) × 𝑣) = ((𝑉 × 𝑉) × 𝑉))
13 oveq12 6558 . . . . . . . . 9 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 WalkOn 𝑒) = (𝑉 WalkOn 𝐸))
1413oveqd 6566 . . . . . . . 8 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑎(𝑣 WalkOn 𝑒)𝑏) = (𝑎(𝑉 WalkOn 𝐸)𝑏))
1514breqd 4594 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝))
16153anbi1d 1395 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → ((𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏)) ↔ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))))
17162exbidv 1839 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃𝑓𝑝(𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏)) ↔ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))))
1812, 17rabeqbidv 3168 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))} = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))})
198, 8, 18mpt2eq123dv 6615 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑎𝑣, 𝑏𝑣 ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))
20 df-2wlkonot 26385 . . 3 2WalksOnOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))
2119, 20ovmpt2ga 6688 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}) ∈ V) → (𝑉 2WalksOnOt 𝐸) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))
222, 4, 7, 21syl3anc 1318 1 ((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOnOt 𝐸) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900  Vcvv 3173   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  1c1 9816  2c2 10947  #chash 12979   WalkOn cwlkon 26030   2WalksOnOt c2wlkonot 26382 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-2wlkonot 26385 This theorem is referenced by:  2wlkonot  26392
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