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Theorem 2wot2wont 25662
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 25654 . . . 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 3059 . . . . 5  |-  w  e. 
_V
3 eleq1 2527 . . . . . 6  |-  ( u  =  w  ->  (
u  e.  ( x ( V 2WalksOnOt  E )
y )  <->  w  e.  ( x ( V 2WalksOnOt  E ) y ) ) )
432rexbidv 2919 . . . . 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 3196 . . . 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 271 . . 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 2459 . 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 4327 . 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 2513 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 375    = wceq 1454    e. wcel 1897   {cab 2447   E.wrex 2749   U_ciun 4291  (class class class)co 6314   2WalksOt c2wlkot 25630   2WalksOnOt c2wlkonot 25631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-1st 6819  df-2nd 6820  df-2wlkonot 25634  df-2wlksot 25635
This theorem is referenced by: (None)
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