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Theorem clwlk 24415
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 3115 . 2  |-  ( V  e.  X  ->  V  e.  _V )
2 elex 3115 . 2  |-  ( E  e.  Y  ->  E  e.  _V )
3 df-clwlk 24412 . . . 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 6284 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Walks  e )  =  ( V Walks  E
) )
65breqd 4451 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Walks 
e ) p  <->  f ( V Walks  E ) p ) )
76anbi1d 704 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( v Walks  e ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) )  <-> 
( f ( V Walks 
E ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) ) )
87adantl 466 . . . 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 4503 . . 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 457 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  V  e.  _V )
11 simpr 461 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
12 id 22 . . . 4  |-  ( f ( V Walks  E ) p  ->  f ( V Walks  E ) p )
1312wlkres 24184 . . 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 6405 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V ClWalks  E )  =  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) } )
151, 2, 14syl2an 477 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 184    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106   class class class wbr 4440   {copab 4497   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   0cc0 9481   #chash 12360   Walks cwalk 24160   ClWalks cclwlk 24409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-word 12495  df-wlk 24170  df-clwlk 24412
This theorem is referenced by:  isclwlk0  24416  clwlkswlks  24420  clwlkcompim  24426
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