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.

Step	Hyp	Ref	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 ) ) ) ) = (/) ) } )
2	1	brfvopab 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 )
8	6, 7	2m1wlk 39681	. . . . . . . . . . . 12 |- ( F (1Walks ` G ) P -> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) )
9	5, 8	syl 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 )
11	10	ad5antr 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 ) ) )
15	12, 13, 14	3jca 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 ) ) )
18	11, 15, 16, 17	syl3c 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 )
19	18	exp31 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 ) ) )
20	9, 19	sylanl1 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 ) ) )
21	20	imp31 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 )
22	4, 21	jca 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 ) )
23	22	exp31 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 ) ) ) )
24	23	3impa 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 ) ) ) )
25	24	com12 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 ) ) ) )
26	3, 25	sylbid 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 ) ) ) )
27	2, 26	mpcom 37	. . 3 |- ( F (PathS ` G ) P -> ( ( P ` 0 ) =/= ( P ` ( # ` F ) ) -> ( F (TrailS ` G ) P /\ Fun `' P ) ) )
28	27	imp 435	. 2 |- ( ( F (PathS ` G ) P /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> ( F (TrailS ` G ) P /\ Fun `' P ) )
29	2	adantr 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 ) ) )
31	29, 30	syl 17	. 2 |- ( ( F (PathS ` G ) P /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> ( F (SPathS ` G ) P <-> ( F (TrailS ` G ) P /\ Fun `' P ) ) )
32	28, 31	mpbird 240	1 |- ( ( F (PathS ` G ) P /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> F (SPathS ` G ) P )

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)