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Theorem spthispth 25301
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 25237 . . 3  |- SPaths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' p ) } )
21brovmpt2ex 6980 . 2  |-  ( F ( V SPaths  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 simprl 762 . . . . 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 5669 . . . . . . 7  |-  ( Fun  `' P  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
54adantl 467 . . . . . 6  |-  ( ( F ( V Trails  E
) P  /\  Fun  `' P )  ->  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) ) )
65adantl 467 . . . . 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 25267 . . . . . . . . . 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 25247 . . . . . . . . 9  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
108, 9syl6 34 . . . . . . . 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 430 . . . . . . 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 5677 . . . . . . . . . . 11  |-  ( Fun  `' P  ->  ( P
" ( { 0 ,  ( # `  F
) }  i^i  (
1..^ ( # `  F
) ) ) )  =  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) ) )
1312adantl 467 . . . . . . . . . 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 10143 . . . . . . . . . . . . . . . . . 18  |-  0  <  1
15 0re 9650 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR
16 1re 9649 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR
1715, 16ltnlei 9762 . . . . . . . . . . . . . . . . . 18  |-  ( 0  <  1  <->  -.  1  <_  0 )
1814, 17mpbi 211 . . . . . . . . . . . . . . . . 17  |-  -.  1  <_  0
19 elfzole1 11935 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  ( 1..^ (
# `  F )
)  ->  1  <_  0 )
2018, 19mto 179 . . . . . . . . . . . . . . . 16  |-  -.  0  e.  ( 1..^ ( # `  F ) )
2120a1i 11 . . . . . . . . . . . . . . 15  |-  ( F  e. Word  dom  E  ->  -.  0  e.  ( 1..^ ( # `  F
) ) )
22 wrdfin 12690 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e. Word  dom  E  ->  F  e.  Fin )
23 hashcl 12544 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e.  Fin  ->  ( # `
 F )  e. 
NN0 )
2423nn0red 10933 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e.  Fin  ->  ( # `
 F )  e.  RR )
2522, 24syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  RR )
2625ltnrd 9776 . . . . . . . . . . . . . . . . 17  |-  ( F  e. Word  dom  E  ->  -.  ( # `  F
)  <  ( # `  F
) )
2726intn3an3d 1376 . . . . . . . . . . . . . . . 16  |-  ( F  e. Word  dom  E  ->  -.  ( ( # `  F
)  e.  ( ZZ>= ` 
1 )  /\  ( # `
 F )  e.  ZZ  /\  ( # `  F )  <  ( # `
 F ) ) )
28 elfzo2 11930 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  ( 1..^ ( # `  F ) )  <->  ( ( # `
 F )  e.  ( ZZ>= `  1 )  /\  ( # `  F
)  e.  ZZ  /\  ( # `  F )  <  ( # `  F
) ) )
2927, 28sylnibr 306 . . . . . . . . . . . . . . 15  |-  ( F  e. Word  dom  E  ->  -.  ( # `  F
)  e.  ( 1..^ ( # `  F
) ) )
30 fvex 5891 . . . . . . . . . . . . . . . 16  |-  ( # `  F )  e.  _V
31 eleq1 2495 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  0  ->  (
x  e.  ( 1..^ ( # `  F
) )  <->  0  e.  ( 1..^ ( # `  F
) ) ) )
3231notbid 295 . . . . . . . . . . . . . . . . 17  |-  ( x  =  0  ->  ( -.  x  e.  (
1..^ ( # `  F
) )  <->  -.  0  e.  ( 1..^ ( # `  F ) ) ) )
33 eleq1 2495 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( # `  F
)  ->  ( x  e.  ( 1..^ ( # `  F ) )  <->  ( # `  F
)  e.  ( 1..^ ( # `  F
) ) ) )
3433notbid 295 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( # `  F
)  ->  ( -.  x  e.  ( 1..^ ( # `  F
) )  <->  -.  ( # `
 F )  e.  ( 1..^ ( # `  F ) ) ) )
3532, 34ralprg 4049 . . . . . . . . . . . . . . . 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 676 . . . . . . . . . . . . . . 15  |-  ( A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) )  <->  ( -.  0  e.  ( 1..^ ( # `  F
) )  /\  -.  ( # `  F )  e.  ( 1..^ (
# `  F )
) ) )
3721, 29, 36sylanbrc 668 . . . . . . . . . . . . . 14  |-  ( F  e. Word  dom  E  ->  A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) ) )
38 disj 3835 . . . . . . . . . . . . . 14  |-  ( ( { 0 ,  (
# `  F ) }  i^i  ( 1..^ (
# `  F )
) )  =  (/)  <->  A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) ) )
3937, 38sylibr 215 . . . . . . . . . . . . 13  |-  ( F  e. Word  dom  E  ->  ( { 0 ,  (
# `  F ) }  i^i  ( 1..^ (
# `  F )
) )  =  (/) )
4039imaeq2d 5187 . . . . . . . . . . . 12  |-  ( F  e. Word  dom  E  ->  ( P " ( { 0 ,  ( # `  F ) }  i^i  ( 1..^ ( # `  F
) ) ) )  =  ( P " (/) ) )
41 ima0 5202 . . . . . . . . . . . 12  |-  ( P
" (/) )  =  (/)
4240, 41syl6eq 2479 . . . . . . . . . . 11  |-  ( F  e. Word  dom  E  ->  ( P " ( { 0 ,  ( # `  F ) }  i^i  ( 1..^ ( # `  F
) ) ) )  =  (/) )
4342adantr 466 . . . . . . . . . 10  |-  ( ( F  e. Word  dom  E  /\  Fun  `' P )  ->  ( P "
( { 0 ,  ( # `  F
) }  i^i  (
1..^ ( # `  F
) ) ) )  =  (/) )
4413, 43eqtr3d 2465 . . . . . . . . 9  |-  ( ( F  e. Word  dom  E  /\  Fun  `' P )  ->  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )
4544ex 435 . . . . . . . 8  |-  ( F  e. Word  dom  E  ->  ( Fun  `' P  -> 
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) )
4645adantr 466 . . . . . . 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 623 . . . . 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 1185 . . . 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 435 . . 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 25297 . . 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 25296 . . 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 271 . 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 37 1  |-  ( F ( V SPaths  E ) P  ->  F ( V Paths  E ) P )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   _Vcvv 3080    i^i cin 3435   (/)c0 3761   {cpr 4000   class class class wbr 4423   `'ccnv 4852   dom cdm 4853    |` cres 4855   "cima 4856   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305   Fincfn 7580   RRcr 9545   0cc0 9546   1c1 9547    < clt 9682    <_ cle 9683   ZZcz 10944   ZZ>=cuz 11166   ...cfz 11791  ..^cfzo 11922   #chash 12521  Word cword 12660   Walks cwalk 25224   Trails ctrail 25225   Paths cpath 25226   SPaths cspath 25227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12522  df-word 12668  df-wlk 25234  df-trail 25235  df-pth 25236  df-spth 25237
This theorem is referenced by:  spthon  25310  isspthonpth  25312  el2spthonot  25596  usg2wotspth  25610  usgra2pthspth  39284  spthdifv  39285  usgra2pth  39287
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