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Theorem 2wot2wont 25012
Description: The set of (simple) paths of length 2 (in a graph) is the set of (simple) paths of length 2 between any two different vertices. (Contributed by Alexander van der Vekens, 27-Feb-2018.)
Assertion
Ref Expression
2wot2wont  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V 2WalksOt  E )  =  U_ x  e.  V  U_ y  e.  V  ( x ( V 2WalksOnOt  E ) y ) )
Distinct variable groups:    x, E, y    x, V, y
Allowed substitution hints:    X( x, y)    Y( x, y)

Proof of Theorem 2wot2wont
Dummy variables  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 el2wlksoton 25004 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( w  e.  ( V 2WalksOt  E )  <->  E. x  e.  V  E. y  e.  V  w  e.  ( x ( V 2WalksOnOt  E ) y ) ) )
2 vex 3112 . . . . 5  |-  w  e. 
_V
3 eleq1 2529 . . . . . 6  |-  ( u  =  w  ->  (
u  e.  ( x ( V 2WalksOnOt  E )
y )  <->  w  e.  ( x ( V 2WalksOnOt  E ) y ) ) )
432rexbidv 2975 . . . . 5  |-  ( u  =  w  ->  ( E. x  e.  V  E. y  e.  V  u  e.  ( x
( V 2WalksOnOt  E ) y )  <->  E. x  e.  V  E. y  e.  V  w  e.  ( x
( V 2WalksOnOt  E ) y ) ) )
52, 4elab 3246 . . . 4  |-  ( w  e.  { u  |  E. x  e.  V  E. y  e.  V  u  e.  ( x
( V 2WalksOnOt  E ) y ) }  <->  E. x  e.  V  E. y  e.  V  w  e.  ( x ( V 2WalksOnOt  E ) y ) )
61, 5syl6bbr 263 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( w  e.  ( V 2WalksOt  E )  <->  w  e.  { u  |  E. x  e.  V  E. y  e.  V  u  e.  ( x ( V 2WalksOnOt  E ) y ) } ) )
76eqrdv 2454 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V 2WalksOt  E )  =  { u  |  E. x  e.  V  E. y  e.  V  u  e.  ( x ( V 2WalksOnOt  E ) y ) } )
8 dfiunv2 4368 . 2  |-  U_ x  e.  V  U_ y  e.  V  ( x ( V 2WalksOnOt  E ) y )  =  { u  |  E. x  e.  V  E. y  e.  V  u  e.  ( x
( V 2WalksOnOt  E ) y ) }
97, 8syl6eqr 2516 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V 2WalksOt  E )  =  U_ x  e.  V  U_ y  e.  V  ( x ( V 2WalksOnOt  E ) y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   E.wrex 2808   U_ciun 4332  (class class class)co 6296   2WalksOt c2wlkot 24980   2WalksOnOt c2wlkonot 24981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-2wlkonot 24984  df-2wlksot 24985
This theorem is referenced by: (None)
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