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Theorem pthdepissPth 39767
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.) (Revised by AV, 12-Jan-2021.)
Assertion
Ref Expression
pthdepissPth  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  F (SPathS `  G ) P )

Proof of Theorem pthdepissPth
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pthsfval 39755 . . . . 5  |-  ( G  e.  _V  ->  (PathS `  G )  =  { <. x ,  y >.  |  ( x (TrailS `  G ) y  /\  Fun  `' ( y  |`  ( 1..^ ( # `  x
) ) )  /\  ( ( y " { 0 ,  (
# `  x ) } )  i^i  (
y " ( 1..^ ( # `  x
) ) ) )  =  (/) ) } )
21brfvopab 6363 . . . 4  |-  ( F (PathS `  G ) P  ->  ( G  e. 
_V  /\  F  e.  _V  /\  P  e.  _V ) )
3 isPth 39757 . . . . 5  |-  ( ( G  e.  _V  /\  F  e.  _V  /\  P  e.  _V )  ->  ( F (PathS `  G ) P 
<->  ( F (TrailS `  G ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
4 simp-4l 781 . . . . . . . . 9  |-  ( ( ( ( ( F (TrailS `  G ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  F
(TrailS `  G ) P )
5 trlis1wlk 39739 . . . . . . . . . . . 12  |-  ( F (TrailS `  G ) P  ->  F (1Walks `  G ) P )
6 eqid 2462 . . . . . . . . . . . . 13  |-  (Vtx `  G )  =  (Vtx
`  G )
7 eqid 2462 . . . . . . . . . . . . 13  |-  (iEdg `  G )  =  (iEdg `  G )
86, 72m1wlk 39681 . . . . . . . . . . . 12  |-  ( F (1Walks `  G ) P  ->  ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G ) ) )
95, 8syl 17 . . . . . . . . . . 11  |-  ( F (TrailS `  G ) P  ->  ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G ) ) )
10 lencl 12722 . . . . . . . . . . . . . 14  |-  ( F  e. Word  dom  (iEdg `  G
)  ->  ( # `  F
)  e.  NN0 )
1110ad5antr 745 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G ) )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  ( # `
 F )  e. 
NN0 )
12 simp-5r 784 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G ) )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G ) )
13 simp-4r 782 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G ) )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) ) )
14 simpr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G ) )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )
1512, 13, 143jca 1194 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G ) )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  ( P : ( 0 ... ( # `  F
) ) --> (Vtx `  G )  /\  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) ) )
16 simpllr 774 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G ) )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  (
( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  (/) )
17 injresinj 12057 . . . . . . . . . . . . 13  |-  ( (
# `  F )  e.  NN0  ->  ( ( P : ( 0 ... ( # `  F
) ) --> (Vtx `  G )  /\  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  (
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/)  ->  Fun  `' P
) ) )
1811, 15, 16, 17syl3c 63 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G ) )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  Fun  `' P )
1918exp31 613 . . . . . . . . . . 11  |-  ( ( ( ( F  e. Word  dom  (iEdg `  G )  /\  P : ( 0 ... ( # `  F
) ) --> (Vtx `  G ) )  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  ->  (
( G  e.  _V  /\  F  e.  _V  /\  P  e.  _V )  ->  ( ( P ` 
0 )  =/=  ( P `  ( # `  F
) )  ->  Fun  `' P ) ) )
209, 19sylanl1 660 . . . . . . . . . 10  |-  ( ( ( F (TrailS `  G ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  ->  (
( G  e.  _V  /\  F  e.  _V  /\  P  e.  _V )  ->  ( ( P ` 
0 )  =/=  ( P `  ( # `  F
) )  ->  Fun  `' P ) ) )
2120imp31 438 . . . . . . . . 9  |-  ( ( ( ( ( F (TrailS `  G ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  Fun  `' P )
224, 21jca 539 . . . . . . . 8  |-  ( ( ( ( ( F (TrailS `  G ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  /\  ( G  e.  _V  /\  F  e.  _V  /\  P  e. 
_V ) )  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  ( F (TrailS `  G ) P  /\  Fun  `' P
) )
2322exp31 613 . . . . . . 7  |-  ( ( ( F (TrailS `  G ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )  /\  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )  ->  (
( G  e.  _V  /\  F  e.  _V  /\  P  e.  _V )  ->  ( ( P ` 
0 )  =/=  ( P `  ( # `  F
) )  ->  ( F (TrailS `  G ) P  /\  Fun  `' P
) ) ) )
24233impa 1210 . . . . . 6  |-  ( ( F (TrailS `  G
) P  /\  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )  ->  ( ( G  e.  _V  /\  F  e.  _V  /\  P  e.  _V )  ->  (
( P `  0
)  =/=  ( P `
 ( # `  F
) )  ->  ( F (TrailS `  G ) P  /\  Fun  `' P
) ) ) )
2524com12 32 . . . . 5  |-  ( ( G  e.  _V  /\  F  e.  _V  /\  P  e.  _V )  ->  (
( F (TrailS `  G ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )  ->  ( ( P `  0 )  =/=  ( P `  ( # `  F ) )  ->  ( F
(TrailS `  G ) P  /\  Fun  `' P
) ) ) )
263, 25sylbid 223 . . . 4  |-  ( ( G  e.  _V  /\  F  e.  _V  /\  P  e.  _V )  ->  ( F (PathS `  G ) P  ->  ( ( P `
 0 )  =/=  ( P `  ( # `
 F ) )  ->  ( F (TrailS `  G ) P  /\  Fun  `' P ) ) ) )
272, 26mpcom 37 . . 3  |-  ( F (PathS `  G ) P  ->  ( ( P `
 0 )  =/=  ( P `  ( # `
 F ) )  ->  ( F (TrailS `  G ) P  /\  Fun  `' P ) ) )
2827imp 435 . 2  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( F
(TrailS `  G ) P  /\  Fun  `' P
) )
292adantr 471 . . 3  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( G  e.  _V  /\  F  e. 
_V  /\  P  e.  _V ) )
30 issPth 39758 . . 3  |-  ( ( G  e.  _V  /\  F  e.  _V  /\  P  e.  _V )  ->  ( F (SPathS `  G ) P 
<->  ( F (TrailS `  G ) P  /\  Fun  `' P ) ) )
3129, 30syl 17 . 2  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  ( F
(SPathS `  G ) P 
<->  ( F (TrailS `  G ) P  /\  Fun  `' P ) ) )
3228, 31mpbird 240 1  |-  ( ( F (PathS `  G
) P  /\  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) )  ->  F (SPathS `  G ) 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  Vtxcvtx 39147  iEdgciedg 39148  1Walksc1wlks 39661  TrailSctrls 39730  PathScpths 39747  SPathScspths 39748
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-ifp 1436  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-1wlks 39665  df-trls 39732  df-pths 39751  df-spths 39752
This theorem is referenced by: (None)
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