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Theorem 0trl 24210
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 4570 . . 3  |-  (/)  e.  _V
2 istrl 24201 . . 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 3925 . . . . 5  |-  A. k  e.  (/)  ( E `  ( (/) `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
5 hash0 12392 . . . . . . . 8  |-  ( # `  (/) )  =  0
65oveq2i 6286 . . . . . . 7  |-  ( 0..^ ( # `  (/) ) )  =  ( 0..^ 0 )
7 fzo0 11806 . . . . . . 7  |-  ( 0..^ 0 )  =  (/)
86, 7eqtri 2489 . . . . . 6  |-  ( 0..^ ( # `  (/) ) )  =  (/)
98raleqi 3055 . . . . 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 2473 . . . . . 6  |-  0  =  ( # `  (/) )
1312oveq2i 6286 . . . . 5  |-  ( 0 ... 0 )  =  ( 0 ... ( # `
 (/) ) )
1413feq2i 5715 . . . 4  |-  ( P : ( 0 ... 0 ) --> V  <->  P :
( 0 ... ( # `
 (/) ) ) --> V )
15 wrd0 12518 . . . . . 6  |-  (/)  e. Word  dom  E
16 fun0 5636 . . . . . . 7  |-  Fun  (/)
17 cnv0 5400 . . . . . . . 8  |-  `' (/)  =  (/)
1817funeqi 5599 . . . . . . 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 970 . . 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 968    = wceq 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106   (/)c0 3778   {cpr 4022   class class class wbr 4440   `'ccnv 4991   dom cdm 4992   Fun wfun 5573   -->wf 5575   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    + caddc 9484   ...cfz 11661  ..^cfzo 11781   #chash 12360  Word cword 12487   Trails ctrail 24161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-wlk 24170  df-trail 24171
This theorem is referenced by:  0trlon  24212  0pth  24234  0spth  24235  0crct  24288
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