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Theorem isclwlkg 30544
Description: Generalisation of isclwlk0 30543: 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, 24-Jun-2018.)
Assertion
Ref Expression
isclwlkg  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( F ( V ClWalks  E ) P  <->  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )

Proof of Theorem isclwlkg
Dummy variables  e 
f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-clwlk 30539 . . . 4  |- ClWalks  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
21brovmpt2ex 6827 . . 3  |-  ( F ( V ClWalks  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 isclwlk0 30543 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V ClWalks  E ) P 
<->  ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
43biimpd 207 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V ClWalks  E ) P  ->  ( F ( V Walks  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )
52, 4mpcom 36 . 2  |-  ( F ( V ClWalks  E ) P  ->  ( F ( V Walks  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) )
6 wlkbprop 23554 . . . . 5  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
76adantr 465 . . . 4  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  (
( # `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
83biimprd 223 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  F
( V ClWalks  E ) P ) )
983adant1 1006 . . . 4  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  F
( V ClWalks  E ) P ) )
107, 9mpcom 36 . . 3  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  F
( V ClWalks  E ) P )
1110a1i 11 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( F ( V Walks  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) )  ->  F ( V ClWalks  E ) P ) )
125, 11impbid2 204 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( 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    /\ w3a 965    = wceq 1370    e. wcel 1757   _Vcvv 3054   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   0cc0 9369   NN0cn0 10666   #chash 12190   Walks cwalk 23526   ClWalks cclwlk 30536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-map 7302  df-pm 7303  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-card 8196  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-fzo 11636  df-hash 12191  df-word 12317  df-wlk 23536  df-clwlk 30539
This theorem is referenced by:  isclwlk  30545
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