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Theorem 2spotfi 26419
 Description: In a finite graph, the set of simple paths of length 2 between two vertices (as ordered triples) is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.)
Assertion
Ref Expression
2spotfi (((𝑉 ∈ Fin ∧ 𝐸𝑋) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) ∈ Fin)

Proof of Theorem 2spotfi
Dummy variables 𝑓 𝑝 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2spthonot 26393 . 2 (((𝑉 ∈ Fin ∧ 𝐸𝑋) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})
2 3xpfi 8117 . . . 4 (𝑉 ∈ Fin → ((𝑉 × 𝑉) × 𝑉) ∈ Fin)
3 rabfi 8070 . . . 4 (((𝑉 × 𝑉) × 𝑉) ∈ Fin → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))} ∈ Fin)
42, 3syl 17 . . 3 (𝑉 ∈ Fin → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))} ∈ Fin)
54ad2antrr 758 . 2 (((𝑉 ∈ Fin ∧ 𝐸𝑋) ∧ (𝐴𝑉𝐵𝑉)) → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))} ∈ Fin)
61, 5eqeltrd 2688 1 (((𝑉 ∈ Fin ∧ 𝐸𝑋) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  1c1 9816  2c2 10947  #chash 12979   SPathOn cspthon 26033   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-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-3or 1032  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-2spthonot 26387 This theorem is referenced by:  frghash2spot  26590
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