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Theorem 0trl 21546
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 4339 . . 3  |-  (/)  e.  _V
2 istrl 21537 . . 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 665 . 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 3732 . . . . 5  |-  A. k  e.  (/)  ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
5 hash0 11646 . . . . . . . 8  |-  ( # `  (/) )  =  0
65oveq2i 6092 . . . . . . 7  |-  ( 0..^ ( # `  (/) ) )  =  ( 0..^ 0 )
7 fzo0 11159 . . . . . . 7  |-  ( 0..^ 0 )  =  (/)
86, 7eqtri 2456 . . . . . 6  |-  ( 0..^ ( # `  (/) ) )  =  (/)
98raleqi 2908 . . . . 5  |-  ( A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  A. k  e.  (/)  ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
104, 9mpbir 201 . . . 4  |-  A. k  e.  ( 0..^ ( # `  (/) ) ) ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
1110biantru 492 . . 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 2440 . . . . . 6  |-  0  =  ( # `  (/) )
1312oveq2i 6092 . . . . 5  |-  ( 0 ... 0 )  =  ( 0 ... ( # `
 (/) ) )
1413feq2i 5586 . . . 4  |-  ( P : ( 0 ... 0 ) --> V  <->  P :
( 0 ... ( # `
 (/) ) ) --> V )
15 wrd0 11732 . . . . . 6  |-  (/)  e. Word  dom  E
16 fun0 5508 . . . . . . 7  |-  Fun  (/)
17 cnv0 5275 . . . . . . . 8  |-  `' (/)  =  (/)
1817funeqi 5474 . . . . . . 7  |-  ( Fun  `' (/)  <->  Fun  (/) )
1916, 18mpbir 201 . . . . . 6  |-  Fun  `' (/)
2015, 19pm3.2i 442 . . . . 5  |-  ( (/)  e. Word  dom  E  /\  Fun  `' (/) )
2120biantrur 493 . . . 4  |-  ( P : ( 0 ... ( # `  (/) ) ) --> V  <->  ( ( (/)  e. Word  dom  E  /\  Fun  `' (/) )  /\  P :
( 0 ... ( # `
 (/) ) ) --> V ) )
2214, 21bitri 241 . . 3  |-  ( P : ( 0 ... 0 ) --> V  <->  ( ( (/) 
e. Word  dom  E  /\  Fun  `' (/) )  /\  P :
( 0 ... ( # `
 (/) ) ) --> V ) )
23 df-3an 938 . . 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 270 . 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 253 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Trails  E ) P  <->  P : ( 0 ... 0 ) --> V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   (/)c0 3628   {cpr 3815   class class class wbr 4212   `'ccnv 4877   dom cdm 4878   Fun wfun 5448   -->wf 5450   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991    + caddc 8993   ...cfz 11043  ..^cfzo 11135   #chash 11618  Word cword 11717   Trails ctrail 21507
This theorem is referenced by:  0trlon  21548  0pth  21570  0spth  21571  0crct  21613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-wlk 21516  df-trail 21517
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