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Theorem spthispth 24992
Description: A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
Assertion
Ref Expression
spthispth  |-  ( F ( V SPaths  E ) P  ->  F ( V Paths  E ) P )

Proof of Theorem spthispth
Dummy variables  x  e  f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-spth 24928 . . 3  |- SPaths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' p ) } )
21brovmpt2ex 6954 . 2  |-  ( F ( V SPaths  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 simprl 756 . . . . 5  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  ( F ( V Trails  E
) P  /\  Fun  `' P ) )  ->  F ( V Trails  E
) P )
4 funres11 5637 . . . . . . 7  |-  ( Fun  `' P  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
54adantl 464 . . . . . 6  |-  ( ( F ( V Trails  E
) P  /\  Fun  `' P )  ->  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) ) )
65adantl 464 . . . . 5  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  ( F ( V Trails  E
) P  /\  Fun  `' P ) )  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
7 trliswlk 24958 . . . . . . . . . 10  |-  ( F ( V Trails  E ) P  ->  F ( V Walks  E ) P )
87a1i 11 . . . . . . . . 9  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Trails  E ) P  ->  F ( V Walks 
E ) P ) )
9 2mwlk 24938 . . . . . . . . 9  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
108, 9syl6 31 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Trails  E ) P  ->  ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V ) ) )
1110imp 427 . . . . . . 7  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  F ( V Trails  E
) P )  -> 
( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V ) )
12 imain 5645 . . . . . . . . . . 11  |-  ( Fun  `' P  ->  ( P
" ( { 0 ,  ( # `  F
) }  i^i  (
1..^ ( # `  F
) ) ) )  =  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) ) )
1312adantl 464 . . . . . . . . . 10  |-  ( ( F  e. Word  dom  E  /\  Fun  `' P )  ->  ( P "
( { 0 ,  ( # `  F
) }  i^i  (
1..^ ( # `  F
) ) ) )  =  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) ) )
14 0lt1 10115 . . . . . . . . . . . . . . . . . 18  |-  0  <  1
15 0re 9626 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR
16 1re 9625 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR
1715, 16ltnlei 9737 . . . . . . . . . . . . . . . . . 18  |-  ( 0  <  1  <->  -.  1  <_  0 )
1814, 17mpbi 208 . . . . . . . . . . . . . . . . 17  |-  -.  1  <_  0
19 elfzole1 11867 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  ( 1..^ (
# `  F )
)  ->  1  <_  0 )
2018, 19mto 176 . . . . . . . . . . . . . . . 16  |-  -.  0  e.  ( 1..^ ( # `  F ) )
2120a1i 11 . . . . . . . . . . . . . . 15  |-  ( F  e. Word  dom  E  ->  -.  0  e.  ( 1..^ ( # `  F
) ) )
22 wrdfin 12613 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e. Word  dom  E  ->  F  e.  Fin )
23 hashcl 12475 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e.  Fin  ->  ( # `
 F )  e. 
NN0 )
2423nn0red 10894 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e.  Fin  ->  ( # `
 F )  e.  RR )
2522, 24syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  RR )
2625ltnrd 9751 . . . . . . . . . . . . . . . . 17  |-  ( F  e. Word  dom  E  ->  -.  ( # `  F
)  <  ( # `  F
) )
2726intn3an3d 1342 . . . . . . . . . . . . . . . 16  |-  ( F  e. Word  dom  E  ->  -.  ( ( # `  F
)  e.  ( ZZ>= ` 
1 )  /\  ( # `
 F )  e.  ZZ  /\  ( # `  F )  <  ( # `
 F ) ) )
28 elfzo2 11862 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  ( 1..^ ( # `  F ) )  <->  ( ( # `
 F )  e.  ( ZZ>= `  1 )  /\  ( # `  F
)  e.  ZZ  /\  ( # `  F )  <  ( # `  F
) ) )
2927, 28sylnibr 303 . . . . . . . . . . . . . . 15  |-  ( F  e. Word  dom  E  ->  -.  ( # `  F
)  e.  ( 1..^ ( # `  F
) ) )
30 fvex 5859 . . . . . . . . . . . . . . . 16  |-  ( # `  F )  e.  _V
31 eleq1 2474 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  0  ->  (
x  e.  ( 1..^ ( # `  F
) )  <->  0  e.  ( 1..^ ( # `  F
) ) ) )
3231notbid 292 . . . . . . . . . . . . . . . . 17  |-  ( x  =  0  ->  ( -.  x  e.  (
1..^ ( # `  F
) )  <->  -.  0  e.  ( 1..^ ( # `  F ) ) ) )
33 eleq1 2474 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( # `  F
)  ->  ( x  e.  ( 1..^ ( # `  F ) )  <->  ( # `  F
)  e.  ( 1..^ ( # `  F
) ) ) )
3433notbid 292 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( # `  F
)  ->  ( -.  x  e.  ( 1..^ ( # `  F
) )  <->  -.  ( # `
 F )  e.  ( 1..^ ( # `  F ) ) ) )
3532, 34ralprg 4021 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR  /\  ( # `  F )  e.  _V )  -> 
( A. x  e. 
{ 0 ,  (
# `  F ) }  -.  x  e.  ( 1..^ ( # `  F
) )  <->  ( -.  0  e.  ( 1..^ ( # `  F
) )  /\  -.  ( # `  F )  e.  ( 1..^ (
# `  F )
) ) ) )
3615, 30, 35mp2an 670 . . . . . . . . . . . . . . 15  |-  ( A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) )  <->  ( -.  0  e.  ( 1..^ ( # `  F
) )  /\  -.  ( # `  F )  e.  ( 1..^ (
# `  F )
) ) )
3721, 29, 36sylanbrc 662 . . . . . . . . . . . . . 14  |-  ( F  e. Word  dom  E  ->  A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) ) )
38 disj 3810 . . . . . . . . . . . . . 14  |-  ( ( { 0 ,  (
# `  F ) }  i^i  ( 1..^ (
# `  F )
) )  =  (/)  <->  A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) ) )
3937, 38sylibr 212 . . . . . . . . . . . . 13  |-  ( F  e. Word  dom  E  ->  ( { 0 ,  (
# `  F ) }  i^i  ( 1..^ (
# `  F )
) )  =  (/) )
4039imaeq2d 5157 . . . . . . . . . . . 12  |-  ( F  e. Word  dom  E  ->  ( P " ( { 0 ,  ( # `  F ) }  i^i  ( 1..^ ( # `  F
) ) ) )  =  ( P " (/) ) )
41 ima0 5172 . . . . . . . . . . . 12  |-  ( P
" (/) )  =  (/)
4240, 41syl6eq 2459 . . . . . . . . . . 11  |-  ( F  e. Word  dom  E  ->  ( P " ( { 0 ,  ( # `  F ) }  i^i  ( 1..^ ( # `  F
) ) ) )  =  (/) )
4342adantr 463 . . . . . . . . . 10  |-  ( ( F  e. Word  dom  E  /\  Fun  `' P )  ->  ( P "
( { 0 ,  ( # `  F
) }  i^i  (
1..^ ( # `  F
) ) ) )  =  (/) )
4413, 43eqtr3d 2445 . . . . . . . . 9  |-  ( ( F  e. Word  dom  E  /\  Fun  `' P )  ->  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )
4544ex 432 . . . . . . . 8  |-  ( F  e. Word  dom  E  ->  ( Fun  `' P  -> 
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) )
4645adantr 463 . . . . . . 7  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( Fun  `' P  ->  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) ) )
4711, 46syl 17 . . . . . 6  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  F ( V Trails  E
) P )  -> 
( Fun  `' P  ->  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) )
4847impr 617 . . . . 5  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  ( F ( V Trails  E
) P  /\  Fun  `' P ) )  -> 
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )
493, 6, 483jca 1177 . . . 4  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  ( F ( V Trails  E
) P  /\  Fun  `' P ) )  -> 
( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) )
5049ex 432 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( F ( V Trails  E
) P  /\  Fun  `' P )  ->  ( F ( V Trails  E
) P  /\  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
51 isspth 24988 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V SPaths  E ) P 
<->  ( F ( V Trails  E ) P  /\  Fun  `' P ) ) )
52 ispth 24987 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Paths  E ) P 
<->  ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
5350, 51, 523imtr4d 268 . 2  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V SPaths  E ) P  ->  F ( V Paths 
E ) P ) )
542, 53mpcom 34 1  |-  ( F ( V SPaths  E ) P  ->  F ( V Paths  E ) P )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   _Vcvv 3059    i^i cin 3413   (/)c0 3738   {cpr 3974   class class class wbr 4395   `'ccnv 4822   dom cdm 4823    |` cres 4825   "cima 4826   Fun wfun 5563   -->wf 5565   ` cfv 5569  (class class class)co 6278   Fincfn 7554   RRcr 9521   0cc0 9522   1c1 9523    < clt 9658    <_ cle 9659   ZZcz 10905   ZZ>=cuz 11127   ...cfz 11726  ..^cfzo 11854   #chash 12452  Word cword 12583   Walks cwalk 24915   Trails ctrail 24916   Paths cpath 24917   SPaths cspath 24918
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
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 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-fzo 11855  df-hash 12453  df-word 12591  df-wlk 24925  df-trail 24926  df-pth 24927  df-spth 24928
This theorem is referenced by:  spthon  25001  isspthonpth  25003  el2spthonot  25287  usg2wotspth  25301  usgra2pthspth  37980  spthdifv  37981  usgra2pth  37983
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