Theorem spthispth 24553

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 24489	. . 3 |- SPaths = ( v e. _V , e e. _V |-> { <. f , p >. | ( f ( v Trails e ) p /\ Fun `' p ) } )
2	1	brovmpt2ex 6953	. 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 5646	. . . . . . 7 |- ( Fun `' P -> Fun `' ( P |` ( 1..^ ( # ` F ) ) ) )
5	4	adantl 466	. . . . . 6 |- ( ( F ( V Trails E ) P /\ Fun `' P ) -> Fun `' ( P |` ( 1..^ ( # ` F ) ) ) )
6	5	adantl 466	. . . . 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 24519	. . . . . . . . . 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 24499	. . . . . . . . 9 |- ( F ( V Walks E ) P -> ( F e. Word dom E /\ P : ( 0 ... ( # ` F ) ) --> V ) )
10	8, 9	syl6 33	. . . . . . . 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 429	. . . . . . 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 5654	. . . . . . . . . . 11 |- ( Fun `' P -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) )
13	12	adantl 466	. . . . . . . . . 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 10082	. . . . . . . . . . . . . . . . . 18 |- 0 < 1
15		0re 9599	. . . . . . . . . . . . . . . . . . 19 |- 0 e. RR
16		1re 9598	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
17	15, 16	ltnlei 9708	. . . . . . . . . . . . . . . . . 18 |- ( 0 < 1 <-> -. 1 <_ 0 )
18	14, 17	mpbi 208	. . . . . . . . . . . . . . . . 17 |- -. 1 <_ 0
19		elfzole1 11818	. . . . . . . . . . . . . . . . 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 12543	. . . . . . . . . . . . . . . . . . 19 |- ( F e. Word dom E -> F e. Fin )
23		hashcl 12410	. . . . . . . . . . . . . . . . . . . 20 |- ( F e. Fin -> ( # ` F ) e. NN0 )
24	23	nn0red 10860	. . . . . . . . . . . . . . . . . . 19 |- ( F e. Fin -> ( # ` F ) e. RR )
25	22, 24	syl 16	. . . . . . . . . . . . . . . . . 18 |- ( F e. Word dom E -> ( # ` F ) e. RR )
26	25	ltnrd 9722	. . . . . . . . . . . . . . . . 17 |- ( F e. Word dom E -> -. ( # ` F ) < ( # ` F ) )
27	26	intn3an3d 1341	. . . . . . . . . . . . . . . 16 |- ( F e. Word dom E -> -. ( ( # ` F ) e. ( ZZ>= ` 1 ) /\ ( # ` F ) e. ZZ /\ ( # ` F ) < ( # ` F ) ) )
28		elfzo2 11814	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) e. ( 1..^ ( # ` F ) ) <-> ( ( # ` F ) e. ( ZZ>= ` 1 ) /\ ( # ` F ) e. ZZ /\ ( # ` F ) < ( # ` F ) ) )
29	27, 28	sylnibr 305	. . . . . . . . . . . . . . 15 |- ( F e. Word dom E -> -. ( # ` F ) e. ( 1..^ ( # ` F ) ) )
30		fvex 5866	. . . . . . . . . . . . . . . 16 |- ( # ` F ) e. _V
31		eleq1 2515	. . . . . . . . . . . . . . . . . 18 |- ( x = 0 -> ( x e. ( 1..^ ( # ` F ) ) <-> 0 e. ( 1..^ ( # ` F ) ) ) )
32	31	notbid 294	. . . . . . . . . . . . . . . . 17 |- ( x = 0 -> ( -. x e. ( 1..^ ( # ` F ) ) <-> -. 0 e. ( 1..^ ( # ` F ) ) ) )
33		eleq1 2515	. . . . . . . . . . . . . . . . . 18 |- ( x = ( # ` F ) -> ( x e. ( 1..^ ( # ` F ) ) <-> ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
34	33	notbid 294	. . . . . . . . . . . . . . . . 17 |- ( x = ( # ` F ) -> ( -. x e. ( 1..^ ( # ` F ) ) <-> -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
35	32, 34	ralprg 4063	. . . . . . . . . . . . . . . 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 672	. . . . . . . . . . . . . . 15 |- ( A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) <-> ( -. 0 e. ( 1..^ ( # ` F ) ) /\ -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
37	21, 29, 36	sylanbrc 664	. . . . . . . . . . . . . 14 |- ( F e. Word dom E -> A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) )
38		disj 3853	. . . . . . . . . . . . . 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 5327	. . . . . . . . . . . 12 |- ( F e. Word dom E -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = ( P " (/) ) )
41		ima0 5342	. . . . . . . . . . . 12 |- ( P " (/) ) = (/)
42	40, 41	syl6eq 2500	. . . . . . . . . . 11 |- ( F e. Word dom E -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = (/) )
43	42	adantr 465	. . . . . . . . . 10 |- ( ( F e. Word dom E /\ Fun `' P ) -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = (/) )
44	13, 43	eqtr3d 2486	. . . . . . . . 9 |- ( ( F e. Word dom E /\ Fun `' P ) -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) )
45	44	ex 434	. . . . . . . 8 |- ( F e. Word dom E -> ( Fun `' P -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) )
46	45	adantr 465	. . . . . . 7 |- ( ( F e. Word dom E /\ P : ( 0 ... ( # ` F ) ) --> V ) -> ( Fun `' P -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) )
47	11, 46	syl 16	. . . . . 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 619	. . . . 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 434	. . 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 24549	. . 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 24548	. . 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 36	1 |- ( F ( V SPaths E ) P -> F ( V Paths E ) P )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   A.wral 2793   _Vcvv 3095   i^i cin 3460   (/)c0 3770   {cpr 4016   class class class wbr 4437   `'ccnv 4988   dom cdm 4989   |` cres 4991   "cima 4992   Fun wfun 5572   -->wf 5574   ` cfv 5578  (class class class)co 6281   Fincfn 7518   RRcr 9494   0cc0 9495   1c1 9496   < clt 9631   <_ cle 9632   ZZcz 10871   ZZ>=cuz 11092   ...cfz 11683  ..^cfzo 11806   #chash 12387  Word cword 12516   Walks cwalk 24476   Trails ctrail 24477   Paths cpath 24478   SPaths cspath 24479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-hash 12388  df-word 12524  df-wlk 24486  df-trail 24487  df-pth 24488  df-spth 24489

This theorem is referenced by:  spthon  24562  isspthonpth  24564  el2spthonot  24848  usg2wotspth  24862  usgra2pthspth  32305  spthdifv  32306  usgra2pth  32308