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Theorem spthispth 24239
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 24175 . . 3  |- SPaths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' p ) } )
21brovmpt2ex 6943 . 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 5649 . . . . . . 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 24205 . . . . . . . . . 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 24185 . . . . . . . . 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 5657 . . . . . . . . . . 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 10066 . . . . . . . . . . . . . . . . . 18  |-  0  <  1
15 0re 9587 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR
16 1re 9586 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR
1715, 16ltnlei 9696 . . . . . . . . . . . . . . . . . 18  |-  ( 0  <  1  <->  -.  1  <_  0 )
1814, 17mpbi 208 . . . . . . . . . . . . . . . . 17  |-  -.  1  <_  0
19 elfzole1 11795 . . . . . . . . . . . . . . . . 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 12516 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e. Word  dom  E  ->  F  e.  Fin )
23 hashcl 12385 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  e.  Fin  ->  ( # `
 F )  e. 
NN0 )
2423nn0red 10844 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e.  Fin  ->  ( # `
 F )  e.  RR )
2522, 24syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  RR )
2625ltnrd 9709 . . . . . . . . . . . . . . . . . 18  |-  ( F  e. Word  dom  E  ->  -.  ( # `  F
)  <  ( # `  F
) )
27 3mix3 1162 . . . . . . . . . . . . . . . . . 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 985 . . . . . . . . . . . . . . . . 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 11791 . . . . . . . . . . . . . . . 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 5869 . . . . . . . . . . . . . . . 16  |-  ( # `  F )  e.  _V
34 eleq1 2534 . . . . . . . . . . . . . . . . . 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 2534 . . . . . . . . . . . . . . . . . 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 4071 . . . . . . . . . . . . . . . 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 3862 . . . . . . . . . . . . . 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 5330 . . . . . . . . . . . 12  |-  ( F  e. Word  dom  E  ->  ( P " ( { 0 ,  ( # `  F ) }  i^i  ( 1..^ ( # `  F
) ) ) )  =  ( P " (/) ) )
44 ima0 5345 . . . . . . . . . . . 12  |-  ( P
" (/) )  =  (/)
4543, 44syl6eq 2519 . . . . . . . . . . 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 2505 . . . . . . . . 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 1171 . . . 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 24235 . . 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 24234 . . 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 967    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   _Vcvv 3108    i^i cin 3470   (/)c0 3780   {cpr 4024   class class class wbr 4442   `'ccnv 4993   dom cdm 4994    |` cres 4996   "cima 4997   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6277   Fincfn 7508   RRcr 9482   0cc0 9483   1c1 9484    < clt 9619    <_ cle 9620   ZZcz 10855   ZZ>=cuz 11073   ...cfz 11663  ..^cfzo 11783   #chash 12362  Word cword 12489   Walks cwalk 24162   Trails ctrail 24163   Paths cpath 24164   SPaths cspath 24165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-fzo 11784  df-hash 12363  df-word 12497  df-wlk 24172  df-trail 24173  df-pth 24174  df-spth 24175
This theorem is referenced by:  spthon  24248  isspthonpth  24250  el2spthonot  24534  usg2wotspth  24548  usgra2pthspth  31777  spthdifv  31778  usgra2pth  31780
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