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Theorem pthdepisspth 24703
Description: A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.)
Assertion
Ref Expression
pthdepisspth  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  F ( V SPaths  E ) P )

Proof of Theorem pthdepisspth
Dummy variables  e 
f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pth 24637 . . . . 5  |- Paths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } )
21brovmpt2ex 6969 . . . 4  |-  ( F ( V Paths  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 ispth 24697 . . . . 5  |-  ( ( ( 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 )
) ) )  =  (/) ) ) )
4 simp-4l 767 . . . . . . . . 9  |-  ( ( ( ( ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  F ( V Trails  E ) P )
5 trliswlk 24668 . . . . . . . . . . . 12  |-  ( F ( V Trails  E ) P  ->  F ( V Walks  E ) P )
6 2mwlk 24648 . . . . . . . . . . . 12  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
75, 6syl 16 . . . . . . . . . . 11  |-  ( F ( V Trails  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
8 lencl 12569 . . . . . . . . . . . . . 14  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
98ad5antr 733 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( # `  F
)  e.  NN0 )
10 simp-5r 770 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  P :
( 0 ... ( # `
 F ) ) --> V )
11 simp-4r 768 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
12 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )
1310, 11, 123jca 1176 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) ) )
14 simpllr 760 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( ( P " { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )
15 injresinj 11929 . . . . . . . . . . . . 13  |-  ( (
# `  F )  e.  NN0  ->  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  (
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/)  ->  Fun  `' P
) ) )
169, 13, 14, 15syl3c 61 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  Fun  `' P
)
1716exp31 604 . . . . . . . . . . 11  |-  ( ( ( ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  -> 
( ( P ` 
0 )  =/=  ( P `  ( # `  F
) )  ->  Fun  `' P ) ) )
187, 17sylanl1 650 . . . . . . . . . 10  |-  ( ( ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  -> 
( ( P ` 
0 )  =/=  ( P `  ( # `  F
) )  ->  Fun  `' P ) ) )
1918imp31 432 . . . . . . . . 9  |-  ( ( ( ( ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  Fun  `' P
)
204, 19jca 532 . . . . . . . 8  |-  ( ( ( ( ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( F
( V Trails  E ) P  /\  Fun  `' P
) )
2120exp31 604 . . . . . . 7  |-  ( ( ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  -> 
( ( P ` 
0 )  =/=  ( P `  ( # `  F
) )  ->  ( F ( V Trails  E
) P  /\  Fun  `' P ) ) ) )
22213impa 1191 . . . . . 6  |-  ( ( F ( V Trails  E
) P  /\  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )  ->  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( P `  0 )  =/=  ( P `  ( # `
 F ) )  ->  ( F ( V Trails  E ) P  /\  Fun  `' P
) ) ) )
2322com12 31 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( F ( V Trails  E
) P  /\  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )  ->  ( ( P `  0 )  =/=  ( P `  ( # `  F ) )  ->  ( F
( V Trails  E ) P  /\  Fun  `' P
) ) ) )
243, 23sylbid 215 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Paths  E ) P  ->  ( ( P `
 0 )  =/=  ( P `  ( # `
 F ) )  ->  ( F ( V Trails  E ) P  /\  Fun  `' P
) ) ) )
252, 24mpcom 36 . . 3  |-  ( F ( V Paths  E ) P  ->  ( ( P `  0 )  =/=  ( P `  ( # `
 F ) )  ->  ( F ( V Trails  E ) P  /\  Fun  `' P
) ) )
2625imp 429 . 2  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( F
( V Trails  E ) P  /\  Fun  `' P
) )
272adantr 465 . . 3  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
28 isspth 24698 . . 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 ) ) )
2927, 28syl 16 . 2  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( F
( V SPaths  E ) P 
<->  ( F ( V Trails  E ) P  /\  Fun  `' P ) ) )
3026, 29mpbird 232 1  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  F ( V SPaths  E ) P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    i^i cin 3470   (/)c0 3793   {cpr 4034   class class class wbr 4456   `'ccnv 5007   dom cdm 5008    |` cres 5010   "cima 5011   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510   NN0cn0 10816   ...cfz 11697  ..^cfzo 11821   #chash 12408  Word cword 12538   Walks cwalk 24625   Trails ctrail 24626   Paths cpath 24627   SPaths cspath 24628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-wlk 24635  df-trail 24636  df-pth 24637  df-spth 24638
This theorem is referenced by: (None)
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