Theorem spthispth 25301

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 25237	. . 3 |- SPaths = ( v e. _V , e e. _V |-> { <. f , p >. | ( f ( v Trails e ) p /\ Fun `' p ) } )
2	1	brovmpt2ex 6980	. 2 |- ( F ( V SPaths E ) P -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )
3		simprl 762	. . . . 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 5669	. . . . . . 7 |- ( Fun `' P -> Fun `' ( P |` ( 1..^ ( # ` F ) ) ) )
5	4	adantl 467	. . . . . 6 |- ( ( F ( V Trails E ) P /\ Fun `' P ) -> Fun `' ( P |` ( 1..^ ( # ` F ) ) ) )
6	5	adantl 467	. . . . 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 25267	. . . . . . . . . 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 25247	. . . . . . . . 9 |- ( F ( V Walks E ) P -> ( F e. Word dom E /\ P : ( 0 ... ( # ` F ) ) --> V ) )
10	8, 9	syl6 34	. . . . . . . 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 430	. . . . . . 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 5677	. . . . . . . . . . 11 |- ( Fun `' P -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) )
13	12	adantl 467	. . . . . . . . . 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 10143	. . . . . . . . . . . . . . . . . 18 |- 0 < 1
15		0re 9650	. . . . . . . . . . . . . . . . . . 19 |- 0 e. RR
16		1re 9649	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
17	15, 16	ltnlei 9762	. . . . . . . . . . . . . . . . . 18 |- ( 0 < 1 <-> -. 1 <_ 0 )
18	14, 17	mpbi 211	. . . . . . . . . . . . . . . . 17 |- -. 1 <_ 0
19		elfzole1 11935	. . . . . . . . . . . . . . . . 17 |- ( 0 e. ( 1..^ ( # ` F ) ) -> 1 <_ 0 )
20	18, 19	mto 179	. . . . . . . . . . . . . . . 16 |- -. 0 e. ( 1..^ ( # ` F ) )
21	20	a1i 11	. . . . . . . . . . . . . . 15 |- ( F e. Word dom E -> -. 0 e. ( 1..^ ( # ` F ) ) )
22		wrdfin 12690	. . . . . . . . . . . . . . . . . . 19 |- ( F e. Word dom E -> F e. Fin )
23		hashcl 12544	. . . . . . . . . . . . . . . . . . . 20 |- ( F e. Fin -> ( # ` F ) e. NN0 )
24	23	nn0red 10933	. . . . . . . . . . . . . . . . . . 19 |- ( F e. Fin -> ( # ` F ) e. RR )
25	22, 24	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( F e. Word dom E -> ( # ` F ) e. RR )
26	25	ltnrd 9776	. . . . . . . . . . . . . . . . 17 |- ( F e. Word dom E -> -. ( # ` F ) < ( # ` F ) )
27	26	intn3an3d 1376	. . . . . . . . . . . . . . . 16 |- ( F e. Word dom E -> -. ( ( # ` F ) e. ( ZZ>= ` 1 ) /\ ( # ` F ) e. ZZ /\ ( # ` F ) < ( # ` F ) ) )
28		elfzo2 11930	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) e. ( 1..^ ( # ` F ) ) <-> ( ( # ` F ) e. ( ZZ>= ` 1 ) /\ ( # ` F ) e. ZZ /\ ( # ` F ) < ( # ` F ) ) )
29	27, 28	sylnibr 306	. . . . . . . . . . . . . . 15 |- ( F e. Word dom E -> -. ( # ` F ) e. ( 1..^ ( # ` F ) ) )
30		fvex 5891	. . . . . . . . . . . . . . . 16 |- ( # ` F ) e. _V
31		eleq1 2495	. . . . . . . . . . . . . . . . . 18 |- ( x = 0 -> ( x e. ( 1..^ ( # ` F ) ) <-> 0 e. ( 1..^ ( # ` F ) ) ) )
32	31	notbid 295	. . . . . . . . . . . . . . . . 17 |- ( x = 0 -> ( -. x e. ( 1..^ ( # ` F ) ) <-> -. 0 e. ( 1..^ ( # ` F ) ) ) )
33		eleq1 2495	. . . . . . . . . . . . . . . . . 18 |- ( x = ( # ` F ) -> ( x e. ( 1..^ ( # ` F ) ) <-> ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
34	33	notbid 295	. . . . . . . . . . . . . . . . 17 |- ( x = ( # ` F ) -> ( -. x e. ( 1..^ ( # ` F ) ) <-> -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
35	32, 34	ralprg 4049	. . . . . . . . . . . . . . . 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 676	. . . . . . . . . . . . . . 15 |- ( A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) <-> ( -. 0 e. ( 1..^ ( # ` F ) ) /\ -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
37	21, 29, 36	sylanbrc 668	. . . . . . . . . . . . . 14 |- ( F e. Word dom E -> A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) )
38		disj 3835	. . . . . . . . . . . . . 14 |- ( ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) = (/) <-> A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) )
39	37, 38	sylibr 215	. . . . . . . . . . . . 13 |- ( F e. Word dom E -> ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) = (/) )
40	39	imaeq2d 5187	. . . . . . . . . . . 12 |- ( F e. Word dom E -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = ( P " (/) ) )
41		ima0 5202	. . . . . . . . . . . 12 |- ( P " (/) ) = (/)
42	40, 41	syl6eq 2479	. . . . . . . . . . 11 |- ( F e. Word dom E -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = (/) )
43	42	adantr 466	. . . . . . . . . 10 |- ( ( F e. Word dom E /\ Fun `' P ) -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = (/) )
44	13, 43	eqtr3d 2465	. . . . . . . . 9 |- ( ( F e. Word dom E /\ Fun `' P ) -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) )
45	44	ex 435	. . . . . . . 8 |- ( F e. Word dom E -> ( Fun `' P -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) )
46	45	adantr 466	. . . . . . 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 623	. . . . 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 1185	. . . 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 435	. . 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 25297	. . 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 25296	. . 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 271	. 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 37	1 |- ( F ( V SPaths E ) P -> F ( V Paths E ) P )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1872   A.wral 2771   _Vcvv 3080   i^i cin 3435   (/)c0 3761   {cpr 4000   class class class wbr 4423   `'ccnv 4852   dom cdm 4853   |` cres 4855   "cima 4856   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305   Fincfn 7580   RRcr 9545   0cc0 9546   1c1 9547   < clt 9682   <_ cle 9683   ZZcz 10944   ZZ>=cuz 11166   ...cfz 11791  ..^cfzo 11922   #chash 12521  Word cword 12660   Walks cwalk 25224   Trails ctrail 25225   Paths cpath 25226   SPaths cspath 25227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  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 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12522  df-word 12668  df-wlk 25234  df-trail 25235  df-pth 25236  df-spth 25237

This theorem is referenced by:  spthon  25310  isspthonpth  25312  el2spthonot  25596  usg2wotspth  25610  usgra2pthspth  39284  spthdifv  39285  usgra2pth  39287