MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0trl Structured version   Unicode version

Theorem 0trl 25262
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 4553 . . 3  |-  (/)  e.  _V
2 istrl 25253 . . 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 687 . 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 3902 . . . . 5  |-  A. k  e.  (/)  ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
5 hash0 12548 . . . . . . . 8  |-  ( # `  (/) )  =  0
65oveq2i 6313 . . . . . . 7  |-  ( 0..^ ( # `  (/) ) )  =  ( 0..^ 0 )
7 fzo0 11943 . . . . . . 7  |-  ( 0..^ 0 )  =  (/)
86, 7eqtri 2451 . . . . . 6  |-  ( 0..^ ( # `  (/) ) )  =  (/)
98raleqi 3029 . . . . 5  |-  ( A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  A. k  e.  (/)  ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
104, 9mpbir 212 . . . 4  |-  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
1110biantru 507 . . 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 2435 . . . . . 6  |-  0  =  ( # `  (/) )
1312oveq2i 6313 . . . . 5  |-  ( 0 ... 0 )  =  ( 0 ... ( # `
 (/) ) )
1413feq2i 5736 . . . 4  |-  ( P : ( 0 ... 0 ) --> V  <->  P :
( 0 ... ( # `
 (/) ) ) --> V )
15 wrd0 12684 . . . . . 6  |-  (/)  e. Word  dom  E
16 fun0 5655 . . . . . . 7  |-  Fun  (/)
17 cnv0 5255 . . . . . . . 8  |-  `' (/)  =  (/)
1817funeqi 5618 . . . . . . 7  |-  ( Fun  `' (/)  <->  Fun  (/) )
1916, 18mpbir 212 . . . . . 6  |-  Fun  `' (/)
2015, 19pm3.2i 456 . . . . 5  |-  ( (/)  e. Word  dom  E  /\  Fun  `' (/) )
2120biantrur 508 . . . 4  |-  ( P : ( 0 ... ( # `  (/) ) ) --> V  <->  ( ( (/)  e. Word  dom  E  /\  Fun  `' (/) )  /\  P :
( 0 ... ( # `
 (/) ) ) --> V ) )
2214, 21bitri 252 . . 3  |-  ( P : ( 0 ... 0 ) --> V  <->  ( ( (/) 
e. Word  dom  E  /\  Fun  `' (/) )  /\  P :
( 0 ... ( # `
 (/) ) ) --> V ) )
23 df-3an 984 . . 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 281 . 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 264 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   _Vcvv 3081   (/)c0 3761   {cpr 3998   class class class wbr 4420   `'ccnv 4849   dom cdm 4850   Fun wfun 5592   -->wf 5594   ` cfv 5598  (class class class)co 6302   0cc0 9540   1c1 9541    + caddc 9543   ...cfz 11785  ..^cfzo 11916   #chash 12515  Word cword 12649   Trails ctrail 25213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-fzo 11917  df-hash 12516  df-word 12657  df-wlk 25222  df-trail 25223
This theorem is referenced by:  0trlon  25264  0pth  25286  0spth  25287  0crct  25340
  Copyright terms: Public domain W3C validator