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Theorem 0wlk 23443
Description: A pair of an empty set (of edges) and a second set (of vertices) is a walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Assertion
Ref Expression
0wlk  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Walks  E ) P  <->  P : ( 0 ... 0 ) --> V ) )

Proof of Theorem 0wlk
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 0ex 4421 . . 3  |-  (/)  e.  _V
2 iswlk 23425 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( (/)  e.  _V  /\  P  e.  Z ) )  ->  ( (/) ( V Walks 
E ) P  <->  ( (/)  e. Word  dom  E  /\  P : ( 0 ... ( # `  (/) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
31, 2mpanr1 683 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Walks  E ) P  <->  ( (/)  e. Word  dom  E  /\  P : ( 0 ... ( # `  (/) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
4 ral0 3783 . . . . 5  |-  A. k  e.  (/)  ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
5 hash0 12134 . . . . . . . 8  |-  ( # `  (/) )  =  0
65oveq2i 6101 . . . . . . 7  |-  ( 0..^ ( # `  (/) ) )  =  ( 0..^ 0 )
7 fzo0 11572 . . . . . . 7  |-  ( 0..^ 0 )  =  (/)
86, 7eqtri 2462 . . . . . 6  |-  ( 0..^ ( # `  (/) ) )  =  (/)
98raleqi 2920 . . . . 5  |-  ( A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  A. k  e.  (/)  ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
104, 9mpbir 209 . . . 4  |-  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
1110biantru 505 . . 3  |-  ( (
(/)  e. Word  dom  E  /\  P : ( 0 ... ( # `  (/) ) ) --> V )  <->  ( ( (/) 
e. Word  dom  E  /\  P : ( 0 ... ( # `  (/) ) ) --> V )  /\  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
125eqcomi 2446 . . . . . 6  |-  0  =  ( # `  (/) )
1312oveq2i 6101 . . . . 5  |-  ( 0 ... 0 )  =  ( 0 ... ( # `
 (/) ) )
1413feq2i 5551 . . . 4  |-  ( P : ( 0 ... 0 ) --> V  <->  P :
( 0 ... ( # `
 (/) ) ) --> V )
15 wrd0 12251 . . . . 5  |-  (/)  e. Word  dom  E
1615biantrur 506 . . . 4  |-  ( P : ( 0 ... ( # `  (/) ) ) --> V  <->  ( (/)  e. Word  dom  E  /\  P : ( 0 ... ( # `  (/) ) ) --> V ) )
1714, 16bitri 249 . . 3  |-  ( P : ( 0 ... 0 ) --> V  <->  ( (/)  e. Word  dom  E  /\  P : ( 0 ... ( # `  (/) ) ) --> V ) )
18 df-3an 967 . . 3  |-  ( (
(/)  e. Word  dom  E  /\  P : ( 0 ... ( # `  (/) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  <->  ( ( (/) 
e. Word  dom  E  /\  P : ( 0 ... ( # `  (/) ) ) --> V )  /\  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
1911, 17, 183bitr4ri 278 . 2  |-  ( (
(/)  e. Word  dom  E  /\  P : ( 0 ... ( # `  (/) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  <->  P :
( 0 ... 0
) --> V )
203, 19syl6bb 261 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Walks  E ) P  <->  P : ( 0 ... 0 ) --> V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   _Vcvv 2971   (/)c0 3636   {cpr 3878   class class class wbr 4291   dom cdm 4839   -->wf 5413   ` cfv 5417  (class class class)co 6090   0cc0 9281   1c1 9282    + caddc 9284   ...cfz 11436  ..^cfzo 11547   #chash 12102  Word cword 12220   Walks cwalk 23404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-fzo 11548  df-hash 12103  df-word 12228  df-wlk 23414
This theorem is referenced by:  0wlkon  23445  0clwlk  30426
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