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Theorem 2spthsot 26395
 Description: The set of simple paths of length 2 (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 28-Feb-2018.)
Assertion
Ref Expression
2spthsot ((𝑉𝑋𝐸𝑌) → (𝑉 2SPathsOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)})
Distinct variable groups:   𝐸,𝑎,𝑏,𝑡   𝑉,𝑎,𝑏,𝑡
Allowed substitution hints:   𝑋(𝑡,𝑎,𝑏)   𝑌(𝑡,𝑎,𝑏)

Proof of Theorem 2spthsot
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 3xpexg 6859 . . . 4 (𝑉𝑋 → ((𝑉 × 𝑉) × 𝑉) ∈ V)
65adantr 480 . . 3 ((𝑉𝑋𝐸𝑌) → ((𝑉 × 𝑉) × 𝑉) ∈ V)
7 rabexg 4739 . . 3 (((𝑉 × 𝑉) × 𝑉) ∈ V → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)} ∈ V)
86, 7syl 17 . 2 ((𝑉𝑋𝐸𝑌) → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)} ∈ V)
9 id 22 . . . . . . 7 (𝑣 = 𝑉𝑣 = 𝑉)
109, 9xpeq12d 5064 . . . . . 6 (𝑣 = 𝑉 → (𝑣 × 𝑣) = (𝑉 × 𝑉))
1110, 9xpeq12d 5064 . . . . 5 (𝑣 = 𝑉 → ((𝑣 × 𝑣) × 𝑣) = ((𝑉 × 𝑉) × 𝑉))
1211adantr 480 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → ((𝑣 × 𝑣) × 𝑣) = ((𝑉 × 𝑉) × 𝑉))
13 simpl 472 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
14 oveq12 6558 . . . . . . . 8 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 2SPathOnOt 𝑒) = (𝑉 2SPathOnOt 𝐸))
1514oveqd 6566 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑎(𝑣 2SPathOnOt 𝑒)𝑏) = (𝑎(𝑉 2SPathOnOt 𝐸)𝑏))
1615eleq2d 2673 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑡 ∈ (𝑎(𝑣 2SPathOnOt 𝑒)𝑏) ↔ 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)))
1713, 16rexeqbidv 3130 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃𝑏𝑣 𝑡 ∈ (𝑎(𝑣 2SPathOnOt 𝑒)𝑏) ↔ ∃𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)))
1813, 17rexeqbidv 3130 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃𝑎𝑣𝑏𝑣 𝑡 ∈ (𝑎(𝑣 2SPathOnOt 𝑒)𝑏) ↔ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)))
1912, 18rabeqbidv 3168 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎𝑣𝑏𝑣 𝑡 ∈ (𝑎(𝑣 2SPathOnOt 𝑒)𝑏)} = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)})
20 df-2spthsot 26388 . . 3 2SPathsOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎𝑣𝑏𝑣 𝑡 ∈ (𝑎(𝑣 2SPathOnOt 𝑒)𝑏)})
2119, 20ovmpt2ga 6688 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)} ∈ V) → (𝑉 2SPathsOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)})
222, 4, 8, 21syl3anc 1318 1 ((𝑉𝑋𝐸𝑌) → (𝑉 2SPathsOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   × cxp 5036  (class class class)co 6549   2SPathsOt c2spthot 26383   2SPathOnOt c2pthonot 26384 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-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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-2spthsot 26388 This theorem is referenced by:  el2spthsoton  26406  2spot0  26591
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