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Theorem isclwlk0 24881
Description: Properties of a pair of functions to be a closed walk (in an undirected graph) in terms of walks. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
Assertion
Ref Expression
isclwlk0  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V ClWalks  E ) P  <->  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )

Proof of Theorem isclwlk0
Dummy variables  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4457 . . 3  |-  ( F ( V ClWalks  E ) P 
<-> 
<. F ,  P >.  e.  ( V ClWalks  E )
)
2 clwlk 24880 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ClWalks  E )  =  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) } )
32adantr 465 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( V ClWalks  E )  =  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) } )
43eleq2d 2527 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( <. F ,  P >.  e.  ( V ClWalks  E
)  <->  <. F ,  P >.  e.  { <. f ,  p >.  |  (
f ( V Walks  E
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) } ) )
51, 4syl5bb 257 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V ClWalks  E ) P  <->  <. F ,  P >.  e.  { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) } ) )
6 breq12 4461 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f ( V Walks 
E ) p  <->  F ( V Walks  E ) P ) )
7 fveq1 5871 . . . . . . 7  |-  ( p  =  P  ->  (
p `  0 )  =  ( P ` 
0 ) )
87adantl 466 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p `  0
)  =  ( P `
 0 ) )
9 simpr 461 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  p  =  P )
10 fveq2 5872 . . . . . . . 8  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
1110adantr 465 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( # `  f
)  =  ( # `  F ) )
129, 11fveq12d 5878 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p `  ( # `
 f ) )  =  ( P `  ( # `  F ) ) )
138, 12eqeq12d 2479 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( p ` 
0 )  =  ( p `  ( # `  f ) )  <->  ( P `  0 )  =  ( P `  ( # `
 F ) ) ) )
146, 13anbi12d 710 . . . 4  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) )  <-> 
( F ( V Walks 
E ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
1514opelopabga 4769 . . 3  |-  ( ( F  e.  W  /\  P  e.  Z )  ->  ( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f ( V Walks  E
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) }  <->  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )
1615adantl 466 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f ( V Walks  E
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) }  <->  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )
175, 16bitrd 253 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V ClWalks  E ) 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 1395    e. wcel 1819   <.cop 4038   class class class wbr 4456   {copab 4514   ` cfv 5594  (class class class)co 6296   0cc0 9509   #chash 12408   Walks cwalk 24625   ClWalks cclwlk 24874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-wlk 24635  df-clwlk 24877
This theorem is referenced by:  isclwlkg  24882  clwlkiswlk  24884  0clwlk  24892
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