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Theorem 0clwlk 25063
Description: A pair of an empty set (of edges) and a second set (of vertices) is a closed walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.)
Assertion
Ref Expression
0clwlk  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V ClWalks  E ) P 
<->  P : ( 0 ... 0 ) --> V ) )

Proof of Theorem 0clwlk
StepHypRef Expression
1 0ex 4525 . . 3  |-  (/)  e.  _V
2 isclwlk0 25052 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( (/)  e.  _V  /\  P  e.  Z ) )  ->  ( (/) ( V ClWalks  E ) P  <->  ( (/) ( V Walks 
E ) P  /\  ( P `  0 )  =  ( P `  ( # `  (/) ) ) ) ) )
31, 2mpanr1 681 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V ClWalks  E ) P 
<->  ( (/) ( V Walks  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  (/) ) ) ) ) )
4 0wlk 24845 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Walks  E ) P  <->  P : ( 0 ... 0 ) --> V ) )
54biimpd 207 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Walks  E ) P  ->  P :
( 0 ... 0
) --> V ) )
65adantrd 466 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  (
( (/) ( V Walks  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  (/) ) ) )  ->  P :
( 0 ... 0
) --> V ) )
7 simplr 754 . . . . . . 7  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  /\  (/) ( V Walks 
E ) P )  /\  P : ( 0 ... 0 ) --> V )  ->  (/) ( V Walks 
E ) P )
8 hash0 12392 . . . . . . . . 9  |-  ( # `  (/) )  =  0
98eqcomi 2415 . . . . . . . 8  |-  0  =  ( # `  (/) )
109fveq2i 5808 . . . . . . 7  |-  ( P `
 0 )  =  ( P `  ( # `
 (/) ) )
117, 10jctir 536 . . . . . 6  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  /\  (/) ( V Walks 
E ) P )  /\  P : ( 0 ... 0 ) --> V )  ->  ( (/) ( V Walks  E ) P  /\  ( P `
 0 )  =  ( P `  ( # `
 (/) ) ) ) )
1211exp31 602 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Walks  E ) P  ->  ( P : ( 0 ... 0 ) --> V  -> 
( (/) ( V Walks  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  (/) ) ) ) ) ) )
134, 12sylbird 235 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( P : ( 0 ... 0 ) --> V  -> 
( P : ( 0 ... 0 ) --> V  ->  ( (/) ( V Walks 
E ) P  /\  ( P `  0 )  =  ( P `  ( # `  (/) ) ) ) ) ) )
1413pm2.43d 47 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( P : ( 0 ... 0 ) --> V  -> 
( (/) ( V Walks  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  (/) ) ) ) ) )
156, 14impbid 191 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  (
( (/) ( V Walks  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  (/) ) ) )  <->  P : ( 0 ... 0 ) --> V ) )
163, 15bitrd 253 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V ClWalks  E ) P 
<->  P : ( 0 ... 0 ) --> V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   (/)c0 3737   class class class wbr 4394   -->wf 5521   ` cfv 5525  (class class class)co 6234   0cc0 9442   ...cfz 11643   #chash 12359   Walks cwalk 24796   ClWalks cclwlk 25045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-map 7379  df-pm 7380  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-card 8272  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-n0 10757  df-z 10826  df-uz 11046  df-fz 11644  df-fzo 11768  df-hash 12360  df-word 12498  df-wlk 24806  df-clwlk 25048
This theorem is referenced by: (None)
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