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Theorem 2wot2wont 30405
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 30397 . . . 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 2975 . . . . 5  |-  w  e. 
_V
3 eleq1 2503 . . . . . . 7  |-  ( u  =  w  ->  (
u  e.  ( x ( V 2WalksOnOt  E )
y )  <->  w  e.  ( x ( V 2WalksOnOt  E ) y ) ) )
43rexbidv 2736 . . . . . 6  |-  ( u  =  w  ->  ( E. y  e.  V  u  e.  ( x
( V 2WalksOnOt  E ) y )  <->  E. y  e.  V  w  e.  ( x
( V 2WalksOnOt  E ) y ) ) )
54rexbidv 2736 . . . . 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 ) ) )
62, 5elab 3106 . . . 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 ) )
71, 6syl6bbr 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 ) } ) )
87eqrdv 2441 . 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 ) } )
9 dfiunv2 4206 . 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 ) }
108, 9syl6eqr 2493 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 1369    e. wcel 1756   {cab 2429   E.wrex 2716   U_ciun 4171  (class class class)co 6091   2WalksOt c2wlkot 30373   2WalksOnOt c2wlkonot 30374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-2wlkonot 30377  df-2wlksot 30378
This theorem is referenced by: (None)
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