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Theorem spthispth 23494
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 23440 . . 3  |- SPaths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' p ) } )
21brovmpt2ex 6762 . 2  |-  ( F ( V SPaths  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 simprl 755 . . . . 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 5507 . . . . . . 7  |-  ( Fun  `' P  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
54adantl 466 . . . . . 6  |-  ( ( F ( V Trails  E
) P  /\  Fun  `' P )  ->  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) ) )
65adantl 466 . . . . 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 23460 . . . . . . . . . 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 23449 . . . . . . . . 9  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
108, 9syl6 33 . . . . . . . 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 429 . . . . . . 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 5515 . . . . . . . . . . 11  |-  ( Fun  `' P  ->  ( P
" ( { 0 ,  ( # `  F
) }  i^i  (
1..^ ( # `  F
) ) ) )  =  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) ) )
1312adantl 466 . . . . . . . . . 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 9883 . . . . . . . . . . . . . . . . . 18  |-  0  <  1
15 0re 9407 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR
16 1re 9406 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR
1715, 16ltnlei 9516 . . . . . . . . . . . . . . . . . 18  |-  ( 0  <  1  <->  -.  1  <_  0 )
1814, 17mpbi 208 . . . . . . . . . . . . . . . . 17  |-  -.  1  <_  0
19 elfzole1 11581 . . . . . . . . . . . . . . . . 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 12269 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e. Word  dom  E  ->  F  e.  Fin )
23 hashcl 12147 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  e.  Fin  ->  ( # `
 F )  e. 
NN0 )
2423nn0red 10658 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e.  Fin  ->  ( # `
 F )  e.  RR )
2522, 24syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  RR )
2625ltnrd 9529 . . . . . . . . . . . . . . . . . 18  |-  ( F  e. Word  dom  E  ->  -.  ( # `  F
)  <  ( # `  F
) )
27 3mix3 1159 . . . . . . . . . . . . . . . . . 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 982 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( ( # `  F
)  e.  ( ZZ>= ` 
1 )  /\  ( # `
 F )  e.  ZZ  /\  ( # `  F )  <  ( # `
 F ) )  <-> 
( -.  ( # `  F )  e.  (
ZZ>= `  1 )  \/ 
-.  ( # `  F
)  e.  ZZ  \/  -.  ( # `  F
)  <  ( # `  F
) ) )
3028, 29sylibr 212 . . . . . . . . . . . . . . . 16  |-  ( F  e. Word  dom  E  ->  -.  ( ( # `  F
)  e.  ( ZZ>= ` 
1 )  /\  ( # `
 F )  e.  ZZ  /\  ( # `  F )  <  ( # `
 F ) ) )
31 elfzo2 11577 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  ( 1..^ ( # `  F ) )  <->  ( ( # `
 F )  e.  ( ZZ>= `  1 )  /\  ( # `  F
)  e.  ZZ  /\  ( # `  F )  <  ( # `  F
) ) )
3230, 31sylnibr 305 . . . . . . . . . . . . . . 15  |-  ( F  e. Word  dom  E  ->  -.  ( # `  F
)  e.  ( 1..^ ( # `  F
) ) )
33 fvex 5722 . . . . . . . . . . . . . . . 16  |-  ( # `  F )  e.  _V
34 eleq1 2503 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  0  ->  (
x  e.  ( 1..^ ( # `  F
) )  <->  0  e.  ( 1..^ ( # `  F
) ) ) )
3534notbid 294 . . . . . . . . . . . . . . . . 17  |-  ( x  =  0  ->  ( -.  x  e.  (
1..^ ( # `  F
) )  <->  -.  0  e.  ( 1..^ ( # `  F ) ) ) )
36 eleq1 2503 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( # `  F
)  ->  ( x  e.  ( 1..^ ( # `  F ) )  <->  ( # `  F
)  e.  ( 1..^ ( # `  F
) ) ) )
3736notbid 294 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( # `  F
)  ->  ( -.  x  e.  ( 1..^ ( # `  F
) )  <->  -.  ( # `
 F )  e.  ( 1..^ ( # `  F ) ) ) )
3835, 37ralprg 3946 . . . . . . . . . . . . . . . 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 672 . . . . . . . . . . . . . . 15  |-  ( A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) )  <->  ( -.  0  e.  ( 1..^ ( # `  F
) )  /\  -.  ( # `  F )  e.  ( 1..^ (
# `  F )
) ) )
4021, 32, 39sylanbrc 664 . . . . . . . . . . . . . 14  |-  ( F  e. Word  dom  E  ->  A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) ) )
41 disj 3740 . . . . . . . . . . . . . 14  |-  ( ( { 0 ,  (
# `  F ) }  i^i  ( 1..^ (
# `  F )
) )  =  (/)  <->  A. x  e.  { 0 ,  ( # `  F
) }  -.  x  e.  ( 1..^ ( # `  F ) ) )
4240, 41sylibr 212 . . . . . . . . . . . . 13  |-  ( F  e. Word  dom  E  ->  ( { 0 ,  (
# `  F ) }  i^i  ( 1..^ (
# `  F )
) )  =  (/) )
4342imaeq2d 5190 . . . . . . . . . . . 12  |-  ( F  e. Word  dom  E  ->  ( P " ( { 0 ,  ( # `  F ) }  i^i  ( 1..^ ( # `  F
) ) ) )  =  ( P " (/) ) )
44 ima0 5205 . . . . . . . . . . . 12  |-  ( P
" (/) )  =  (/)
4543, 44syl6eq 2491 . . . . . . . . . . 11  |-  ( F  e. Word  dom  E  ->  ( P " ( { 0 ,  ( # `  F ) }  i^i  ( 1..^ ( # `  F
) ) ) )  =  (/) )
4645adantr 465 . . . . . . . . . 10  |-  ( ( F  e. Word  dom  E  /\  Fun  `' P )  ->  ( P "
( { 0 ,  ( # `  F
) }  i^i  (
1..^ ( # `  F
) ) ) )  =  (/) )
4713, 46eqtr3d 2477 . . . . . . . . 9  |-  ( ( F  e. Word  dom  E  /\  Fun  `' P )  ->  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )
4847ex 434 . . . . . . . 8  |-  ( F  e. Word  dom  E  ->  ( Fun  `' P  -> 
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) )
4948adantr 465 . . . . . . 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 619 . . . . 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 1168 . . . 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 434 . . 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 23490 . . 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 23489 . . 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 268 . 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 36 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 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2736   _Vcvv 2993    i^i cin 3348   (/)c0 3658   {cpr 3900   class class class wbr 4313   `'ccnv 4860   dom cdm 4861    |` cres 4863   "cima 4864   Fun wfun 5433   -->wf 5435   ` cfv 5439  (class class class)co 6112   Fincfn 7331   RRcr 9302   0cc0 9303   1c1 9304    < clt 9439    <_ cle 9440   ZZcz 10667   ZZ>=cuz 10882   ...cfz 11458  ..^cfzo 11569   #chash 12124  Word cword 12242   Walks cwalk 23427   Trails ctrail 23428   Paths cpath 23429   SPaths cspath 23430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-fzo 11570  df-hash 12125  df-word 12250  df-wlk 23437  df-trail 23438  df-pth 23439  df-spth 23440
This theorem is referenced by:  spthon  23503  isspthonpth  23505  usgra2pthspth  30321  spthdifv  30325  usgra2pth  30327  el2spthonot  30415  usg2wotspth  30429
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