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Theorem 0clwlk 30353
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 4419 . . 3  |-  (/)  e.  _V
2 isclwlk0 30344 . . 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 678 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V ClWalks  E ) P 
<->  ( (/) ( V Walks  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  (/) ) ) ) ) )
4 0wlk 23379 . . . . 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 465 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  (
( (/) ( V Walks  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  (/) ) ) )  ->  P :
( 0 ... 0
) --> V ) )
7 simplr 749 . . . . . . 7  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  /\  (/) ( V Walks 
E ) P )  /\  P : ( 0 ... 0 ) --> V )  ->  (/) ( V Walks 
E ) P )
8 hash0 12131 . . . . . . . . 9  |-  ( # `  (/) )  =  0
98eqcomi 2445 . . . . . . . 8  |-  0  =  ( # `  (/) )
109fveq2i 5691 . . . . . . 7  |-  ( P `
 0 )  =  ( P `  ( # `
 (/) ) )
117, 10jctir 535 . . . . . 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 601 . . . . 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 48 . . 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 369    = wceq 1364    e. wcel 1761   _Vcvv 2970   (/)c0 3634   class class class wbr 4289   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278   ...cfz 11433   #chash 12099   Walks cwalk 23340   ClWalks cclwlk 30337
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  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 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-wlk 23350  df-clwlk 30340
This theorem is referenced by: (None)
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