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Theorem isclwlkg 24960
Description: Generalisation of isclwlk0 24959: 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 24955 . . . 4  |- ClWalks  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
21brovmpt2ex 6943 . . 3  |-  ( F ( V ClWalks  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 isclwlk0 24959 . . . 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 24728 . . . . 5  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
76adantr 463 . . . 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 1012 . . . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   0cc0 9481   NN0cn0 10791   #chash 12390   Walks cwalk 24703   ClWalks cclwlk 24952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-wlk 24713  df-clwlk 24955
This theorem is referenced by:  isclwlk  24961
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