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Theorem 0trl 23443
Description: A pair of an empty set (of edges) and a second set (of vertices) is a trail 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
0trl  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Trails  E ) P  <->  P : ( 0 ... 0 ) --> V ) )

Proof of Theorem 0trl
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 0ex 4420 . . 3  |-  (/)  e.  _V
2 istrl 23434 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( (/)  e.  _V  /\  P  e.  Z ) )  ->  ( (/) ( V Trails  E ) P  <->  ( ( (/) 
e. Word  dom  E  /\  Fun  `' (/) )  /\  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 Trails  E ) P  <->  ( ( (/)  e. Word  dom  E  /\  Fun  `' (/) )  /\  P :
( 0 ... ( # `
 (/) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
4 ral0 3782 . . . . 5  |-  A. k  e.  (/)  ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
5 hash0 12133 . . . . . . . 8  |-  ( # `  (/) )  =  0
65oveq2i 6100 . . . . . . 7  |-  ( 0..^ ( # `  (/) ) )  =  ( 0..^ 0 )
7 fzo0 11571 . . . . . . 7  |-  ( 0..^ 0 )  =  (/)
86, 7eqtri 2461 . . . . . 6  |-  ( 0..^ ( # `  (/) ) )  =  (/)
98raleqi 2919 . . . . 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  /\  Fun  `' (/) )  /\  P : ( 0 ... ( # `  (/) ) ) --> V )  <->  ( (
( (/)  e. Word  dom  E  /\  Fun  `' (/) )  /\  P : ( 0 ... ( # `  (/) ) ) --> V )  /\  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
125eqcomi 2445 . . . . . 6  |-  0  =  ( # `  (/) )
1312oveq2i 6100 . . . . 5  |-  ( 0 ... 0 )  =  ( 0 ... ( # `
 (/) ) )
1413feq2i 5550 . . . 4  |-  ( P : ( 0 ... 0 ) --> V  <->  P :
( 0 ... ( # `
 (/) ) ) --> V )
15 wrd0 12250 . . . . . 6  |-  (/)  e. Word  dom  E
16 fun0 5473 . . . . . . 7  |-  Fun  (/)
17 cnv0 5238 . . . . . . . 8  |-  `' (/)  =  (/)
1817funeqi 5436 . . . . . . 7  |-  ( Fun  `' (/)  <->  Fun  (/) )
1916, 18mpbir 209 . . . . . 6  |-  Fun  `' (/)
2015, 19pm3.2i 455 . . . . 5  |-  ( (/)  e. Word  dom  E  /\  Fun  `' (/) )
2120biantrur 506 . . . 4  |-  ( P : ( 0 ... ( # `  (/) ) ) --> V  <->  ( ( (/)  e. Word  dom  E  /\  Fun  `' (/) )  /\  P :
( 0 ... ( # `
 (/) ) ) --> V ) )
2214, 21bitri 249 . . 3  |-  ( P : ( 0 ... 0 ) --> V  <->  ( ( (/) 
e. Word  dom  E  /\  Fun  `' (/) )  /\  P :
( 0 ... ( # `
 (/) ) ) --> V ) )
23 df-3an 967 . . 3  |-  ( ( ( (/)  e. Word  dom  E  /\  Fun  `' (/) )  /\  P : ( 0 ... ( # `  (/) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  <->  ( (
( (/)  e. Word  dom  E  /\  Fun  `' (/) )  /\  P : ( 0 ... ( # `  (/) ) ) --> V )  /\  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
2411, 22, 233bitr4ri 278 . 2  |-  ( ( ( (/)  e. Word  dom  E  /\  Fun  `' (/) )  /\  P : ( 0 ... ( # `  (/) ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  <->  P :
( 0 ... 0
) --> V )
253, 24syl6bb 261 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Trails  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 2713   _Vcvv 2970   (/)c0 3635   {cpr 3877   class class class wbr 4290   `'ccnv 4837   dom cdm 4838   Fun wfun 5410   -->wf 5412   ` cfv 5416  (class class class)co 6089   0cc0 9280   1c1 9281    + caddc 9283   ...cfz 11435  ..^cfzo 11546   #chash 12101  Word cword 12219   Trails ctrail 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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-wlk 23413  df-trail 23414
This theorem is referenced by:  0trlon  23445  0pth  23467  0spth  23468  0crct  23510
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