Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  is2spthonot Structured version   Unicode version

Theorem is2spthonot 25590
 Description: The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
Assertion
Ref Expression
is2spthonot 2SPathOnOt SPathOn
Distinct variable groups:   ,,,,,   ,,,,,
Allowed substitution hints:   (,,,,)   (,,,,)

Proof of Theorem is2spthonot
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3089 . . 3
3 elex 3089 . . 3
5 mpt2exga 6883 . . . 4 SPathOn
65anidms 649 . . 3 SPathOn
8 simpl 458 . . . 4
9 id 22 . . . . . . . 8
109, 9xpeq12d 4878 . . . . . . 7
1110, 9xpeq12d 4878 . . . . . 6
1211adantr 466 . . . . 5
13 oveq12 6314 . . . . . . . . 9 SPathOn SPathOn
1413oveqd 6322 . . . . . . . 8 SPathOn SPathOn
1514breqd 4434 . . . . . . 7 SPathOn SPathOn
16153anbi1d 1339 . . . . . 6 SPathOn SPathOn
17162exbidv 1764 . . . . 5 SPathOn SPathOn
1812, 17rabeqbidv 3075 . . . 4 SPathOn SPathOn
198, 8, 18mpt2eq123dv 6367 . . 3 SPathOn SPathOn
20 df-2spthonot 25586 . . 3 2SPathOnOt SPathOn
2119, 20ovmpt2ga 6440 . 2 SPathOn 2SPathOnOt SPathOn
222, 4, 7, 21syl3anc 1264 1 2SPathOnOt SPathOn
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   w3a 982   wceq 1437  wex 1657   wcel 1872  crab 2775  cvv 3080   class class class wbr 4423   cxp 4851  cfv 5601  (class class class)co 6305   cmpt2 6307  c1st 6805  c2nd 6806  c1 9547  c2 10666  chash 12521   SPathOn cspthon 25231   2SPathOnOt c2pthonot 25583 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-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597 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-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  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-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-2spthonot 25586 This theorem is referenced by:  2spthonot  25592
 Copyright terms: Public domain W3C validator