Theorem spthispth 24992

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.

Step	Hyp	Ref	Expression
1		df-spth 24928	. . 3 |- SPaths = ( v e. _V , e e. _V |-> { <. f , p >. | ( f ( v Trails e ) p /\ Fun `' p ) } )
2	1	brovmpt2ex 6954	. 2 |- ( F ( V SPaths E ) P -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )
3		simprl 756	. . . . 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 5637	. . . . . . 7 |- ( Fun `' P -> Fun `' ( P |` ( 1..^ ( # ` F ) ) ) )
5	4	adantl 464	. . . . . 6 |- ( ( F ( V Trails E ) P /\ Fun `' P ) -> Fun `' ( P |` ( 1..^ ( # ` F ) ) ) )
6	5	adantl 464	. . . . 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 24958	. . . . . . . . . 10 |- ( F ( V Trails E ) P -> F ( V Walks E ) P )
8	7	a1i 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 24938	. . . . . . . . 9 |- ( F ( V Walks E ) P -> ( F e. Word dom E /\ P : ( 0 ... ( # ` F ) ) --> V ) )
10	8, 9	syl6 31	. . . . . . . 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 ) ) )
11	10	imp 427	. . . . . . 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 5645	. . . . . . . . . . 11 |- ( Fun `' P -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) )
13	12	adantl 464	. . . . . . . . . 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 10115	. . . . . . . . . . . . . . . . . 18 |- 0 < 1
15		0re 9626	. . . . . . . . . . . . . . . . . . 19 |- 0 e. RR
16		1re 9625	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
17	15, 16	ltnlei 9737	. . . . . . . . . . . . . . . . . 18 |- ( 0 < 1 <-> -. 1 <_ 0 )
18	14, 17	mpbi 208	. . . . . . . . . . . . . . . . 17 |- -. 1 <_ 0
19		elfzole1 11867	. . . . . . . . . . . . . . . . 17 |- ( 0 e. ( 1..^ ( # ` F ) ) -> 1 <_ 0 )
20	18, 19	mto 176	. . . . . . . . . . . . . . . 16 |- -. 0 e. ( 1..^ ( # ` F ) )
21	20	a1i 11	. . . . . . . . . . . . . . 15 |- ( F e. Word dom E -> -. 0 e. ( 1..^ ( # ` F ) ) )
22		wrdfin 12613	. . . . . . . . . . . . . . . . . . 19 |- ( F e. Word dom E -> F e. Fin )
23		hashcl 12475	. . . . . . . . . . . . . . . . . . . 20 |- ( F e. Fin -> ( # ` F ) e. NN0 )
24	23	nn0red 10894	. . . . . . . . . . . . . . . . . . 19 |- ( F e. Fin -> ( # ` F ) e. RR )
25	22, 24	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( F e. Word dom E -> ( # ` F ) e. RR )
26	25	ltnrd 9751	. . . . . . . . . . . . . . . . 17 |- ( F e. Word dom E -> -. ( # ` F ) < ( # ` F ) )
27	26	intn3an3d 1342	. . . . . . . . . . . . . . . 16 |- ( F e. Word dom E -> -. ( ( # ` F ) e. ( ZZ>= ` 1 ) /\ ( # ` F ) e. ZZ /\ ( # ` F ) < ( # ` F ) ) )
28		elfzo2 11862	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) e. ( 1..^ ( # ` F ) ) <-> ( ( # ` F ) e. ( ZZ>= ` 1 ) /\ ( # ` F ) e. ZZ /\ ( # ` F ) < ( # ` F ) ) )
29	27, 28	sylnibr 303	. . . . . . . . . . . . . . 15 |- ( F e. Word dom E -> -. ( # ` F ) e. ( 1..^ ( # ` F ) ) )
30		fvex 5859	. . . . . . . . . . . . . . . 16 |- ( # ` F ) e. _V
31		eleq1 2474	. . . . . . . . . . . . . . . . . 18 |- ( x = 0 -> ( x e. ( 1..^ ( # ` F ) ) <-> 0 e. ( 1..^ ( # ` F ) ) ) )
32	31	notbid 292	. . . . . . . . . . . . . . . . 17 |- ( x = 0 -> ( -. x e. ( 1..^ ( # ` F ) ) <-> -. 0 e. ( 1..^ ( # ` F ) ) ) )
33		eleq1 2474	. . . . . . . . . . . . . . . . . 18 |- ( x = ( # ` F ) -> ( x e. ( 1..^ ( # ` F ) ) <-> ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
34	33	notbid 292	. . . . . . . . . . . . . . . . 17 |- ( x = ( # ` F ) -> ( -. x e. ( 1..^ ( # ` F ) ) <-> -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
35	32, 34	ralprg 4021	. . . . . . . . . . . . . . . 16 |- ( ( 0 e. RR /\ ( # ` F ) e. _V ) -> ( A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) <-> ( -. 0 e. ( 1..^ ( # ` F ) ) /\ -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) ) )
36	15, 30, 35	mp2an 670	. . . . . . . . . . . . . . 15 |- ( A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) <-> ( -. 0 e. ( 1..^ ( # ` F ) ) /\ -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
37	21, 29, 36	sylanbrc 662	. . . . . . . . . . . . . 14 |- ( F e. Word dom E -> A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) )
38		disj 3810	. . . . . . . . . . . . . 14 |- ( ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) = (/) <-> A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) )
39	37, 38	sylibr 212	. . . . . . . . . . . . 13 |- ( F e. Word dom E -> ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) = (/) )
40	39	imaeq2d 5157	. . . . . . . . . . . 12 |- ( F e. Word dom E -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = ( P " (/) ) )
41		ima0 5172	. . . . . . . . . . . 12 |- ( P " (/) ) = (/)
42	40, 41	syl6eq 2459	. . . . . . . . . . 11 |- ( F e. Word dom E -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = (/) )
43	42	adantr 463	. . . . . . . . . 10 |- ( ( F e. Word dom E /\ Fun `' P ) -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = (/) )
44	13, 43	eqtr3d 2445	. . . . . . . . 9 |- ( ( F e. Word dom E /\ Fun `' P ) -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) )
45	44	ex 432	. . . . . . . 8 |- ( F e. Word dom E -> ( Fun `' P -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) )
46	45	adantr 463	. . . . . . 7 |- ( ( F e. Word dom E /\ P : ( 0 ... ( # ` F ) ) --> V ) -> ( Fun `' P -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) )
47	11, 46	syl 17	. . . . . 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 ) ) ) ) = (/) ) )
48	47	impr 617	. . . . 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 ) ) ) ) = (/) )
49	3, 6, 48	3jca 1177	. . . 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 ) ) ) ) = (/) ) )
50	49	ex 432	. . 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 ) ) ) ) = (/) ) ) )
51		isspth 24988	. . 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 ) ) )
52		ispth 24987	. . 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 ) ) ) ) = (/) ) ) )
53	50, 51, 52	3imtr4d 268	. 2 |- ( ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( V SPaths E ) P -> F ( V Paths E ) P ) )
54	2, 53	mpcom 34	1 |- ( F ( V SPaths E ) P -> F ( V Paths E ) P )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   A.wral 2754   _Vcvv 3059   i^i cin 3413   (/)c0 3738   {cpr 3974   class class class wbr 4395   `'ccnv 4822   dom cdm 4823   |` cres 4825   "cima 4826   Fun wfun 5563   -->wf 5565   ` cfv 5569  (class class class)co 6278   Fincfn 7554   RRcr 9521   0cc0 9522   1c1 9523   < clt 9658   <_ cle 9659   ZZcz 10905   ZZ>=cuz 11127   ...cfz 11726  ..^cfzo 11854   #chash 12452  Word cword 12583   Walks cwalk 24915   Trails ctrail 24916   Paths cpath 24917   SPaths cspath 24918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-fzo 11855  df-hash 12453  df-word 12591  df-wlk 24925  df-trail 24926  df-pth 24927  df-spth 24928

This theorem is referenced by:  spthon  25001  isspthonpth  25003  el2spthonot  25287  usg2wotspth  25301  usgra2pthspth  37980  spthdifv  37981  usgra2pth  37983