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Theorem clwlkswlks 25478
Description: Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.)
Assertion
Ref Expression
clwlkswlks  |-  ( W  e.  ( V ClWalks  E
)  ->  W  e.  ( V Walks  E ) )

Proof of Theorem clwlkswlks
Dummy variables  e 
f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-clwlk 25470 . . 3  |- ClWalks  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
21elmpt2cl 6523 . 2  |-  ( W  e.  ( V ClWalks  E
)  ->  ( V  e.  _V  /\  E  e. 
_V ) )
3 clwlk 25473 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V ClWalks  E )  =  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) } )
43eleq2d 2493 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( W  e.  ( V ClWalks  E )  <->  W  e.  {
<. f ,  p >.  |  ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } ) )
5 simpl 459 . . . . . . 7  |-  ( ( f ( V Walks  E
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) )  ->  f
( V Walks  E )
p )
65a1i 11 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) )  ->  f ( V Walks 
E ) p ) )
76ssopab2dv 4747 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) }  C_  { <. f ,  p >.  |  f
( V Walks  E )
p } )
87sseld 3464 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( W  e.  { <. f ,  p >.  |  ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) }  ->  W  e.  { <. f ,  p >.  |  f ( V Walks  E
) p } ) )
9 elopab 4726 . . . . 5  |-  ( W  e.  { <. f ,  p >.  |  f
( V Walks  E )
p }  <->  E. f E. p ( W  = 
<. f ,  p >.  /\  f ( V Walks  E
) p ) )
10 df-br 4422 . . . . . . . . . 10  |-  ( f ( V Walks  E ) p  <->  <. f ,  p >.  e.  ( V Walks  E
) )
1110biimpi 198 . . . . . . . . 9  |-  ( f ( V Walks  E ) p  ->  <. f ,  p >.  e.  ( V Walks  E ) )
1211adantl 468 . . . . . . . 8  |-  ( ( W  =  <. f ,  p >.  /\  f
( V Walks  E )
p )  ->  <. f ,  p >.  e.  ( V Walks  E ) )
13 eleq1 2495 . . . . . . . . 9  |-  ( W  =  <. f ,  p >.  ->  ( W  e.  ( V Walks  E )  <->  <. f ,  p >.  e.  ( V Walks  E ) ) )
1413adantr 467 . . . . . . . 8  |-  ( ( W  =  <. f ,  p >.  /\  f
( V Walks  E )
p )  ->  ( W  e.  ( V Walks  E )  <->  <. f ,  p >.  e.  ( V Walks  E
) ) )
1512, 14mpbird 236 . . . . . . 7  |-  ( ( W  =  <. f ,  p >.  /\  f
( V Walks  E )
p )  ->  W  e.  ( V Walks  E ) )
1615a1i 11 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( W  = 
<. f ,  p >.  /\  f ( V Walks  E
) p )  ->  W  e.  ( V Walks  E ) ) )
1716exlimdvv 1770 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( E. f E. p ( W  = 
<. f ,  p >.  /\  f ( V Walks  E
) p )  ->  W  e.  ( V Walks  E ) ) )
189, 17syl5bi 221 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( W  e.  { <. f ,  p >.  |  f ( V Walks  E
) p }  ->  W  e.  ( V Walks  E
) ) )
198, 18syld 46 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( W  e.  { <. f ,  p >.  |  ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) }  ->  W  e.  ( V Walks  E ) ) )
204, 19sylbid 219 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( W  e.  ( V ClWalks  E )  ->  W  e.  ( V Walks  E ) ) )
212, 20mpcom 38 1  |-  ( W  e.  ( V ClWalks  E
)  ->  W  e.  ( V Walks  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438   E.wex 1660    e. wcel 1869   _Vcvv 3082   <.cop 4003   class class class wbr 4421   {copab 4479   ` cfv 5599  (class class class)co 6303   0cc0 9541   #chash 12516   Walks cwalk 25218   ClWalks cclwlk 25467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-fzo 11918  df-hash 12517  df-word 12662  df-wlk 25228  df-clwlk 25470
This theorem is referenced by:  clwlksarewlks  25479  clwlkfoclwwlk  25565  clwlkf1clwwlk  25570
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