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Theorem is2wlk 23399
Description: Properties of a pair of functions to be a walk of length 2 (in an undirected graph). (Contributed by Alexander van der Vekens, 16-Feb-2018.)
Assertion
Ref Expression
is2wlk  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( ( F ( V Walks  E ) P  /\  ( # `  F
)  =  2 )  <-> 
( F : ( 0..^ 2 ) --> dom 
E  /\  P :
( 0 ... 2
) --> V  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) ) )

Proof of Theorem is2wlk
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 iswlk 23361 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V Walks 
E ) P  <->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
21anbi1d 699 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( ( F ( V Walks  E ) P  /\  ( # `  F
)  =  2 )  <-> 
( ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } )  /\  ( # `  F )  =  2 ) ) )
3 wrdf 12236 . . . . . . . . . 10  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
4 oveq2 6098 . . . . . . . . . . 11  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
54feq2d 5544 . . . . . . . . . 10  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 2 ) --> dom  E ) )
63, 5syl5ib 219 . . . . . . . . 9  |-  ( (
# `  F )  =  2  ->  ( F  e. Word  dom  E  ->  F : ( 0..^ 2 ) --> dom  E )
)
7 iswrdi 12235 . . . . . . . . 9  |-  ( F : ( 0..^ 2 ) --> dom  E  ->  F  e. Word  dom  E )
86, 7impbid1 203 . . . . . . . 8  |-  ( (
# `  F )  =  2  ->  ( F  e. Word  dom  E  <->  F :
( 0..^ 2 ) --> dom  E ) )
98adantl 463 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
)  /\  ( # `  F
)  =  2 )  ->  ( F  e. Word  dom  E  <->  F : ( 0..^ 2 ) --> dom  E
) )
10 oveq2 6098 . . . . . . . . 9  |-  ( (
# `  F )  =  2  ->  (
0 ... ( # `  F
) )  =  ( 0 ... 2 ) )
1110feq2d 5544 . . . . . . . 8  |-  ( (
# `  F )  =  2  ->  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... 2
) --> V ) )
1211adantl 463 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
)  /\  ( # `  F
)  =  2 )  ->  ( P :
( 0 ... ( # `
 F ) ) --> V  <->  P : ( 0 ... 2 ) --> V ) )
13 fzo0to2pr 11610 . . . . . . . . . . 11  |-  ( 0..^ 2 )  =  {
0 ,  1 }
144, 13syl6eq 2489 . . . . . . . . . 10  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
1514raleqdv 2921 . . . . . . . . 9  |-  ( (
# `  F )  =  2  ->  ( A. k  e.  (
0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  A. k  e.  {
0 ,  1 }  ( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
16 2wlklem 23398 . . . . . . . . 9  |-  ( A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) )
1715, 16syl6bb 261 . . . . . . . 8  |-  ( (
# `  F )  =  2  ->  ( A. k  e.  (
0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) ) )
1817adantl 463 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
)  /\  ( # `  F
)  =  2 )  ->  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  <->  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) )
199, 12, 183anbi123d 1284 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
)  /\  ( # `  F
)  =  2 )  ->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  <->  ( F : ( 0..^ 2 ) --> dom  E  /\  P : ( 0 ... 2 ) --> V  /\  ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) ) )
20 3anass 964 . . . . . 6  |-  ( ( F : ( 0..^ 2 ) --> dom  E  /\  P : ( 0 ... 2 ) --> V  /\  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) )  <->  ( F : ( 0..^ 2 ) --> dom  E  /\  ( P : ( 0 ... 2 ) --> V  /\  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) ) ) )
2119, 20syl6bb 261 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
)  /\  ( # `  F
)  =  2 )  ->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  <->  ( F : ( 0..^ 2 ) --> dom  E  /\  ( P : ( 0 ... 2 ) --> V  /\  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) ) ) ) )
2221ex 434 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( ( # `  F
)  =  2  -> 
( ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } )  <->  ( F : ( 0..^ 2 ) --> dom  E  /\  ( P : ( 0 ... 2 ) --> V  /\  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) ) ) ) ) )
2322pm5.32rd 635 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  /\  ( # `
 F )  =  2 )  <->  ( ( F : ( 0..^ 2 ) --> dom  E  /\  ( P : ( 0 ... 2 ) --> V  /\  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) ) )  /\  ( # `  F
)  =  2 ) ) )
24 3anass 964 . . . 4  |-  ( ( ( F : ( 0..^ 2 ) --> dom 
E  /\  ( # `  F
)  =  2 )  /\  P : ( 0 ... 2 ) --> V  /\  ( ( E `  ( F `
 0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  <-> 
( ( F :
( 0..^ 2 ) --> dom  E  /\  ( # `
 F )  =  2 )  /\  ( P : ( 0 ... 2 ) --> V  /\  ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) ) )
25 an32 791 . . . 4  |-  ( ( ( F : ( 0..^ 2 ) --> dom 
E  /\  ( # `  F
)  =  2 )  /\  ( P :
( 0 ... 2
) --> V  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) )  <->  ( ( F : ( 0..^ 2 ) --> dom  E  /\  ( P : ( 0 ... 2 ) --> V  /\  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) ) )  /\  ( # `  F
)  =  2 ) )
2624, 25bitri 249 . . 3  |-  ( ( ( F : ( 0..^ 2 ) --> dom 
E  /\  ( # `  F
)  =  2 )  /\  P : ( 0 ... 2 ) --> V  /\  ( ( E `  ( F `
 0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  <-> 
( ( F :
( 0..^ 2 ) --> dom  E  /\  ( P : ( 0 ... 2 ) --> V  /\  ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) )  /\  ( # `  F )  =  2 ) )
2723, 26syl6bbr 263 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  /\  ( # `
 F )  =  2 )  <->  ( ( F : ( 0..^ 2 ) --> dom  E  /\  ( # `  F )  =  2 )  /\  P : ( 0 ... 2 ) --> V  /\  ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) ) )
28 ffn 5556 . . . . . . 7  |-  ( F : ( 0..^ 2 ) --> dom  E  ->  F  Fn  ( 0..^ 2 ) )
29 hashfn 12134 . . . . . . . 8  |-  ( F  Fn  ( 0..^ 2 )  ->  ( # `  F
)  =  ( # `  ( 0..^ 2 ) ) )
30 2nn0 10592 . . . . . . . . 9  |-  2  e.  NN0
31 hashfzo0 12187 . . . . . . . . 9  |-  ( 2  e.  NN0  ->  ( # `  ( 0..^ 2 ) )  =  2 )
3230, 31ax-mp 5 . . . . . . . 8  |-  ( # `  ( 0..^ 2 ) )  =  2
3329, 32syl6eq 2489 . . . . . . 7  |-  ( F  Fn  ( 0..^ 2 )  ->  ( # `  F
)  =  2 )
3428, 33syl 16 . . . . . 6  |-  ( F : ( 0..^ 2 ) --> dom  E  ->  (
# `  F )  =  2 )
3534pm4.71i 627 . . . . 5  |-  ( F : ( 0..^ 2 ) --> dom  E  <->  ( F : ( 0..^ 2 ) --> dom  E  /\  ( # `  F )  =  2 ) )
3635bicomi 202 . . . 4  |-  ( ( F : ( 0..^ 2 ) --> dom  E  /\  ( # `  F
)  =  2 )  <-> 
F : ( 0..^ 2 ) --> dom  E
)
3736a1i 11 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( ( F :
( 0..^ 2 ) --> dom  E  /\  ( # `
 F )  =  2 )  <->  F :
( 0..^ 2 ) --> dom  E ) )
38373anbi1d 1288 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( ( ( F : ( 0..^ 2 ) --> dom  E  /\  ( # `  F )  =  2 )  /\  P : ( 0 ... 2 ) --> V  /\  ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  <-> 
( F : ( 0..^ 2 ) --> dom 
E  /\  P :
( 0 ... 2
) --> V  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) ) )
392, 27, 383bitrd 279 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( ( F ( V Walks  E ) P  /\  ( # `  F
)  =  2 )  <-> 
( F : ( 0..^ 2 ) --> dom 
E  /\  P :
( 0 ... 2
) --> V  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   {cpr 3876   class class class wbr 4289   dom cdm 4836    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279    + caddc 9281   2c2 10367   NN0cn0 10575   ...cfz 11433  ..^cfzo 11544   #chash 12099  Word cword 12217   Walks cwalk 23340
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-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-wlk 23350
This theorem is referenced by:  usg2wlkonot  30327  usg2wotspth  30328
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