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Theorem clwlk 25352
Description: The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.)
Assertion
Ref Expression
clwlk  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ClWalks  E )  =  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) } )
Distinct variable groups:    f, E, p    f, V, p
Allowed substitution hints:    X( f, p)    Y( f, p)

Proof of Theorem clwlk
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3087 . 2  |-  ( V  e.  X  ->  V  e.  _V )
2 elex 3087 . 2  |-  ( E  e.  Y  ->  E  e.  _V )
3 df-clwlk 25349 . . . 4  |- ClWalks  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
43a1i 11 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  -> ClWalks 
=  ( v  e. 
_V ,  e  e. 
_V  |->  { <. f ,  p >.  |  (
f ( v Walks  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) } ) )
5 oveq12 6305 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Walks  e )  =  ( V Walks  E
) )
65breqd 4428 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Walks 
e ) p  <->  f ( V Walks  E ) p ) )
76anbi1d 709 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( v Walks  e ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) )  <-> 
( f ( V Walks 
E ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) ) )
87adantl 467 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( f ( v Walks 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) )  <->  ( f
( V Walks  E )
p  /\  ( p `  0 )  =  ( p `  ( # `
 f ) ) ) ) )
98opabbidv 4480 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) }  =  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) } )
10 simpl 458 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  V  e.  _V )
11 simpr 462 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
12 id 23 . . . 4  |-  ( f ( V Walks  E ) p  ->  f ( V Walks  E ) p )
1312wlkres 25121 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) }  e.  _V )
144, 9, 10, 11, 13ovmpt2d 6429 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V ClWalks  E )  =  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) } )
151, 2, 14syl2an 479 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ClWalks  E )  =  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   _Vcvv 3078   class class class wbr 4417   {copab 4474   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   0cc0 9528   #chash 12501   Walks cwalk 25097   ClWalks cclwlk 25346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-fzo 11903  df-hash 12502  df-word 12640  df-wlk 25107  df-clwlk 25349
This theorem is referenced by:  isclwlk0  25353  clwlkswlks  25357  clwlkcompim  25363
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