MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  is2wlk Structured version   Unicode version

Theorem is2wlk 24271
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 24224 . . 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 12519 . . . . . . . . . 10  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
4 oveq2 6292 . . . . . . . . . . 11  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
54feq2d 5718 . . . . . . . . . 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 12518 . . . . . . . . 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 6292 . . . . . . . . 9  |-  ( (
# `  F )  =  2  ->  (
0 ... ( # `  F
) )  =  ( 0 ... 2 ) )
1110feq2d 5718 . . . . . . . 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 11867 . . . . . . . . . . 11  |-  ( 0..^ 2 )  =  {
0 ,  1 }
144, 13syl6eq 2524 . . . . . . . . . 10  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
1514raleqdv 3064 . . . . . . . . 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 24270 . . . . . . . . 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 796 . . . 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 5731 . . . . . . 7  |-  ( F : ( 0..^ 2 ) --> dom  E  ->  F  Fn  ( 0..^ 2 ) )
29 hashfn 12411 . . . . . . . 8  |-  ( F  Fn  ( 0..^ 2 )  ->  ( # `  F
)  =  ( # `  ( 0..^ 2 ) ) )
30 2nn0 10812 . . . . . . . . 9  |-  2  e.  NN0
31 hashfzo0 12453 . . . . . . . . 9  |-  ( 2  e.  NN0  ->  ( # `  ( 0..^ 2 ) )  =  2 )
3230, 31ax-mp 5 . . . . . . . 8  |-  ( # `  ( 0..^ 2 ) )  =  2
3329, 32syl6eq 2524 . . . . . . 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 1379    e. wcel 1767   A.wral 2814   {cpr 4029   class class class wbr 4447   dom cdm 4999    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   0cc0 9492   1c1 9493    + caddc 9495   2c2 10585   NN0cn0 10795   ...cfz 11672  ..^cfzo 11792   #chash 12373  Word cword 12500   Walks cwalk 24202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-wlk 24212
This theorem is referenced by:  usg2wlkonot  24587  usg2wotspth  24588
  Copyright terms: Public domain W3C validator