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Theorem spthispth 21578
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 21524 . . 3  |- SPaths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' p ) } )
21brovmpt2ex 6478 . 2  |-  ( F ( V SPaths  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 simprl 734 . . . . 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 5524 . . . . . . 7  |-  ( Fun  `' P  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
54adantl 454 . . . . . 6  |-  ( ( F ( V Trails  E
) P  /\  Fun  `' P )  ->  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) ) )
65adantl 454 . . . . 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 21544 . . . . . . . . . 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 21533 . . . . . . . . 9  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
108, 9syl6 32 . . . . . . . 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 420 . . . . . . 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 5532 . . . . . . . . . . 11  |-  ( Fun  `' P  ->  ( P
" ( { 0 ,  ( # `  F
) }  i^i  (
1..^ ( # `  F
) ) ) )  =  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) ) )
1312adantl 454 . . . . . . . . . 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 9555 . . . . . . . . . . . . . . . . . 18  |-  0  <  1
15 0re 9096 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR
16 1re 9095 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR
1715, 16ltnlei 9199 . . . . . . . . . . . . . . . . . 18  |-  ( 0  <  1  <->  -.  1  <_  0 )
1814, 17mpbi 201 . . . . . . . . . . . . . . . . 17  |-  -.  1  <_  0
19 elfzole1 11152 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  ( 1..^ (
# `  F )
)  ->  1  <_  0 )
2018, 19mto 170 . . . . . . . . . . . . . . . 16  |-  -.  0  e.  ( 1..^ ( # `  F ) )
2120a1i 11 . . . . . . . . . . . . . . 15  |-  ( F  e. Word  dom  E  ->  -.  0  e.  ( 1..^ ( # `  F
) ) )
22 wrdfin 11739 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e. Word  dom  E  ->  F  e.  Fin )
23 hashcl 11644 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  e.  Fin  ->  ( # `
 F )  e. 
NN0 )
2423nn0red 10280 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e.  Fin  ->  ( # `
 F )  e.  RR )
2522, 24syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  RR )
2625ltnrd 9212 . . . . . . . . . . . . . . . . . 18  |-  ( F  e. Word  dom  E  ->  -.  ( # `  F
)  <  ( # `  F
) )
27 3mix3 1129 . . . . . . . . . . . . . . . . . 18  |-  ( -.  ( # `  F
)  <  ( # `  F
)  ->  ( -.  ( # `  F )  e.  ( ZZ>= `  1
)  \/  -.  ( # `
 F )  e.  ZZ  \/  -.  ( # `
 F )  < 
( # `  F ) ) )
2826, 27syl 16 . . . . . . . . . . . . . . . . 17  |-  ( F  e. Word  dom  E  ->  ( -.  ( # `  F
)  e.  ( ZZ>= ` 
1 )  \/  -.  ( # `  F )  e.  ZZ  \/  -.  ( # `  F )  <  ( # `  F
) ) )
29 3ianor 952 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( ( # `  F
)  e.  ( ZZ>= ` 
1 )  /\  ( # `
 F )  e.  ZZ  /\  ( # `  F )  <  ( # `
 F ) )  <-> 
( -.  ( # `  F )  e.  (
ZZ>= `  1 )  \/ 
-.  ( # `  F
)  e.  ZZ  \/  -.  ( # `  F
)  <  ( # `  F
) ) )
3028, 29sylibr 205 . . . . . . . . . . . . . . . 16  |-  ( F  e. Word  dom  E  ->  -.  ( ( # `  F
)  e.  ( ZZ>= ` 
1 )  /\  ( # `
 F )  e.  ZZ  /\  ( # `  F )  <  ( # `
 F ) ) )
31 elfzo2 11148 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  ( 1..^ ( # `  F ) )  <->  ( ( # `
 F )  e.  ( ZZ>= `  1 )  /\  ( # `  F
)  e.  ZZ  /\  ( # `  F )  <  ( # `  F
) ) )
3230, 31sylnibr 298 . . . . . . . . . . . . . . 15  |-  ( F  e. Word  dom  E  ->  -.  ( # `  F
)  e.  ( 1..^ ( # `  F
) ) )
33 fvex 5745 . . . . . . . . . . . . . . . 16  |-  ( # `  F )  e.  _V
34 eleq1 2498 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  0  ->  (
x  e.  ( 1..^ ( # `  F
) )  <->  0  e.  ( 1..^ ( # `  F
) ) ) )
3534notbid 287 . . . . . . . . . . . . . . . . 17  |-  ( x  =  0  ->  ( -.  x  e.  (
1..^ ( # `  F
) )  <->  -.  0  e.  ( 1..^ ( # `  F ) ) ) )
36 eleq1 2498 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( # `  F
)  ->  ( x  e.  ( 1..^ ( # `  F ) )  <->  ( # `  F
)  e.  ( 1..^ ( # `  F
) ) ) )
3736notbid 287 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( # `  F
)  ->  ( -.  x  e.  ( 1..^ ( # `  F
) )  <->  -.  ( # `
 F )  e.  ( 1..^ ( # `  F ) ) ) )
3835, 37ralprg 3859 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR  /\  ( # `  F )  e.  _V )  -> 
( A. x  e. 
{ 0 ,  (
# `  F ) }  -.  x  e.  ( 1..^ ( # `  F
) )  <->  ( -.  0  e.  ( 1..^ ( # `  F
) )  /\  -.  ( # `  F )  e.  ( 1..^ (
# `  F )
) ) ) )
3915, 33, 38mp2an 655 . . . . . . . . . . . . . . 15  |-  ( A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) )  <->  ( -.  0  e.  ( 1..^ ( # `  F
) )  /\  -.  ( # `  F )  e.  ( 1..^ (
# `  F )
) ) )
4021, 32, 39sylanbrc 647 . . . . . . . . . . . . . 14  |-  ( F  e. Word  dom  E  ->  A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) ) )
41 disj 3670 . . . . . . . . . . . . . 14  |-  ( ( { 0 ,  (
# `  F ) }  i^i  ( 1..^ (
# `  F )
) )  =  (/)  <->  A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) ) )
4240, 41sylibr 205 . . . . . . . . . . . . 13  |-  ( F  e. Word  dom  E  ->  ( { 0 ,  (
# `  F ) }  i^i  ( 1..^ (
# `  F )
) )  =  (/) )
4342imaeq2d 5206 . . . . . . . . . . . 12  |-  ( F  e. Word  dom  E  ->  ( P " ( { 0 ,  ( # `  F ) }  i^i  ( 1..^ ( # `  F
) ) ) )  =  ( P " (/) ) )
44 ima0 5224 . . . . . . . . . . . 12  |-  ( P
" (/) )  =  (/)
4543, 44syl6eq 2486 . . . . . . . . . . 11  |-  ( F  e. Word  dom  E  ->  ( P " ( { 0 ,  ( # `  F ) }  i^i  ( 1..^ ( # `  F
) ) ) )  =  (/) )
4645adantr 453 . . . . . . . . . 10  |-  ( ( F  e. Word  dom  E  /\  Fun  `' P )  ->  ( P "
( { 0 ,  ( # `  F
) }  i^i  (
1..^ ( # `  F
) ) ) )  =  (/) )
4713, 46eqtr3d 2472 . . . . . . . . 9  |-  ( ( F  e. Word  dom  E  /\  Fun  `' P )  ->  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )
4847ex 425 . . . . . . . 8  |-  ( F  e. Word  dom  E  ->  ( Fun  `' P  -> 
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) )
4948adantr 453 . . . . . . 7  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( Fun  `' P  ->  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) ) )
5011, 49syl 16 . . . . . 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 )
) ) )  =  (/) ) )
5150impr 604 . . . . 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 )
) ) )  =  (/) )
523, 6, 513jca 1135 . . . 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 )
) ) )  =  (/) ) )
5352ex 425 . . 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 )
) ) )  =  (/) ) ) )
54 isspth 21574 . . 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 ) ) )
55 ispth 21573 . . 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 )
) ) )  =  (/) ) ) )
5653, 54, 553imtr4d 261 . 2  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V SPaths  E ) P  ->  F ( V Paths 
E ) P ) )
572, 56mpcom 35 1  |-  ( F ( V SPaths  E ) P  ->  F ( V Paths  E ) P )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    \/ w3o 936    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    i^i cin 3321   (/)c0 3630   {cpr 3817   class class class wbr 4215   `'ccnv 4880   dom cdm 4881    |` cres 4883   "cima 4884   Fun wfun 5451   -->wf 5453   ` cfv 5457  (class class class)co 6084   Fincfn 7112   RRcr 8994   0cc0 8995   1c1 8996    < clt 9125    <_ cle 9126   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048  ..^cfzo 11140   #chash 11623  Word cword 11722   Walks cwalk 21511   Trails ctrail 21512   Paths cpath 21513   SPaths cspath 21514
This theorem is referenced by:  spthon  21587  isspthonpth  21589  usgra2pthspth  28342  spthdifv  28346  usgra2pth  28348  el2spthonot  28401  usg2wotspth  28415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141  df-hash 11624  df-word 11728  df-wlk 21521  df-trail 21522  df-pth 21523  df-spth 21524
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