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Theorem 2spthsot 25595
 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 2SPathOnOt 2SPathOnOt
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem 2spthsot
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3089 . . 3
21adantr 466 . 2
3 elex 3089 . . 3
43adantl 467 . 2
5 3xpexg 6609 . . . 4
65adantr 466 . . 3
7 rabexg 4574 . . 3 2SPathOnOt
86, 7syl 17 . 2 2SPathOnOt
9 id 22 . . . . . . 7
109, 9xpeq12d 4878 . . . . . 6
1110, 9xpeq12d 4878 . . . . 5
1211adantr 466 . . . 4
13 simpl 458 . . . . 5
14 oveq12 6315 . . . . . . . 8 2SPathOnOt 2SPathOnOt
1514oveqd 6323 . . . . . . 7 2SPathOnOt 2SPathOnOt
1615eleq2d 2492 . . . . . 6 2SPathOnOt 2SPathOnOt
1713, 16rexeqbidv 3037 . . . . 5 2SPathOnOt 2SPathOnOt
1813, 17rexeqbidv 3037 . . . 4 2SPathOnOt 2SPathOnOt
1912, 18rabeqbidv 3075 . . 3 2SPathOnOt 2SPathOnOt
20 df-2spthsot 25588 . . 3 2SPathOnOt 2SPathOnOt
2119, 20ovmpt2ga 6441 . 2 2SPathOnOt 2SPathOnOt 2SPathOnOt
222, 4, 8, 21syl3anc 1264 1 2SPathOnOt 2SPathOnOt
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   wceq 1437   wcel 1872  wrex 2772  crab 2775  cvv 3080   cxp 4851  (class class class)co 6306   2SPathOnOt c2spthot 25583   2SPathOnOt c2pthonot 25584 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-2spthsot 25588 This theorem is referenced by:  el2spthsoton  25606  2spot0  25791
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