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Theorem pthdepisspth 25353
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 25287 . . . . 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 6996 . . . 4  |-  ( F ( V Paths  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 ispth 25347 . . . . 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 781 . . . . . . . . 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 25318 . . . . . . . . . . . 12  |-  ( F ( V Trails  E ) P  ->  F ( V Walks  E ) P )
6 2mwlk 25298 . . . . . . . . . . . 12  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
75, 6syl 17 . . . . . . . . . . 11  |-  ( F ( V Trails  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
8 lencl 12722 . . . . . . . . . . . . . 14  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
98ad5antr 745 . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . 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 774 . . . . . . . . . . . . 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 12057 . . . . . . . . . . . . 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 63 . . . . . . . . . . . 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 613 . . . . . . . . . . 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 660 . . . . . . . . . 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 438 . . . . . . . . 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 539 . . . . . . . 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 613 . . . . . . 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 1210 . . . . . 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 32 . . . . 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 223 . . . 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 37 . . 3  |-  ( F ( V Paths  E ) P  ->  ( ( P `  0 )  =/=  ( P `  ( # `
 F ) )  ->  ( F ( V Trails  E ) P  /\  Fun  `' P
) ) )
2625imp 435 . 2  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( F
( V Trails  E ) P  /\  Fun  `' P
) )
272adantr 471 . . 3  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
28 isspth 25348 . . 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 17 . 2  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( F
( V SPaths  E ) P 
<->  ( F ( V Trails  E ) P  /\  Fun  `' P ) ) )
3026, 29mpbird 240 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   _Vcvv 3057    i^i cin 3415   (/)c0 3743   {cpr 3982   class class class wbr 4416   `'ccnv 4852   dom cdm 4853    |` cres 4855   "cima 4856   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6315   0cc0 9565   1c1 9566   NN0cn0 10898   ...cfz 11813  ..^cfzo 11946   #chash 12547  Word cword 12689   Walks cwalk 25275   Trails ctrail 25276   Paths cpath 25277   SPaths cspath 25278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  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 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-pm 7501  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-card 8399  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-n0 10899  df-z 10967  df-uz 11189  df-fz 11814  df-fzo 11947  df-hash 12548  df-word 12697  df-wlk 25285  df-trail 25286  df-pth 25287  df-spth 25288
This theorem is referenced by: (None)
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