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Theorem is2wlk 24694
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 24647 . . 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 704 . 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 12558 . . . . . . . . . 10  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
4 oveq2 6304 . . . . . . . . . . 11  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
54feq2d 5724 . . . . . . . . . 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 12557 . . . . . . . . 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 466 . . . . . . 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 6304 . . . . . . . . 9  |-  ( (
# `  F )  =  2  ->  (
0 ... ( # `  F
) )  =  ( 0 ... 2 ) )
1110feq2d 5724 . . . . . . . 8  |-  ( (
# `  F )  =  2  ->  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... 2
) --> V ) )
1211adantl 466 . . . . . . 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 11902 . . . . . . . . . . 11  |-  ( 0..^ 2 )  =  {
0 ,  1 }
144, 13syl6eq 2514 . . . . . . . . . 10  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
1514raleqdv 3060 . . . . . . . . 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 24693 . . . . . . . . 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 466 . . . . . . 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 1299 . . . . . 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 977 . . . . . 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 640 . . 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 977 . . . 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 798 . . . 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 5737 . . . . . . 7  |-  ( F : ( 0..^ 2 ) --> dom  E  ->  F  Fn  ( 0..^ 2 ) )
29 hashfn 12446 . . . . . . . 8  |-  ( F  Fn  ( 0..^ 2 )  ->  ( # `  F
)  =  ( # `  ( 0..^ 2 ) ) )
30 2nn0 10833 . . . . . . . . 9  |-  2  e.  NN0
31 hashfzo0 12492 . . . . . . . . 9  |-  ( 2  e.  NN0  ->  ( # `  ( 0..^ 2 ) )  =  2 )
3230, 31ax-mp 5 . . . . . . . 8  |-  ( # `  ( 0..^ 2 ) )  =  2
3329, 32syl6eq 2514 . . . . . . 7  |-  ( F  Fn  ( 0..^ 2 )  ->  ( # `  F
)  =  2 )
3428, 33syl 16 . . . . . 6  |-  ( F : ( 0..^ 2 ) --> dom  E  ->  (
# `  F )  =  2 )
3534pm4.71i 632 . . . . 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 1303 . 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   {cpr 4034   class class class wbr 4456   dom cdm 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512   2c2 10606   NN0cn0 10816   ...cfz 11697  ..^cfzo 11821   #chash 12408  Word cword 12538   Walks cwalk 24625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-wlk 24635
This theorem is referenced by:  usg2wlkonot  25010  usg2wotspth  25011
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