Theorem spthispth 25382

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 25318	. . 3 |- SPaths = ( v e. _V , e e. _V |-> { <. f , p >. | ( f ( v Trails e ) p /\ Fun `' p ) } )
2	1	brovmpt2ex 6988	. 2 |- ( F ( V SPaths E ) P -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )
3		simprl 772	. . . . 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 5661	. . . . . . 7 |- ( Fun `' P -> Fun `' ( P |` ( 1..^ ( # ` F ) ) ) )
5	4	adantl 473	. . . . . 6 |- ( ( F ( V Trails E ) P /\ Fun `' P ) -> Fun `' ( P |` ( 1..^ ( # ` F ) ) ) )
6	5	adantl 473	. . . . 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 25348	. . . . . . . . . 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 25328	. . . . . . . . 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 436	. . . . . . 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 5669	. . . . . . . . . . 11 |- ( Fun `' P -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) )
13	12	adantl 473	. . . . . . . . . 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 10157	. . . . . . . . . . . . . . . . . 18 |- 0 < 1
15		0re 9661	. . . . . . . . . . . . . . . . . . 19 |- 0 e. RR
16		1re 9660	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
17	15, 16	ltnlei 9773	. . . . . . . . . . . . . . . . . 18 |- ( 0 < 1 <-> -. 1 <_ 0 )
18	14, 17	mpbi 213	. . . . . . . . . . . . . . . . 17 |- -. 1 <_ 0
19		elfzole1 11955	. . . . . . . . . . . . . . . . 17 |- ( 0 e. ( 1..^ ( # ` F ) ) -> 1 <_ 0 )
20	18, 19	mto 181	. . . . . . . . . . . . . . . 16 |- -. 0 e. ( 1..^ ( # ` F ) )
21	20	a1i 11	. . . . . . . . . . . . . . 15 |- ( F e. Word dom E -> -. 0 e. ( 1..^ ( # ` F ) ) )
22		wrdfin 12736	. . . . . . . . . . . . . . . . . . 19 |- ( F e. Word dom E -> F e. Fin )
23		hashcl 12576	. . . . . . . . . . . . . . . . . . . 20 |- ( F e. Fin -> ( # ` F ) e. NN0 )
24	23	nn0red 10950	. . . . . . . . . . . . . . . . . . 19 |- ( F e. Fin -> ( # ` F ) e. RR )
25	22, 24	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( F e. Word dom E -> ( # ` F ) e. RR )
26	25	ltnrd 9786	. . . . . . . . . . . . . . . . 17 |- ( F e. Word dom E -> -. ( # ` F ) < ( # ` F ) )
27	26	intn3an3d 1409	. . . . . . . . . . . . . . . 16 |- ( F e. Word dom E -> -. ( ( # ` F ) e. ( ZZ>= ` 1 ) /\ ( # ` F ) e. ZZ /\ ( # ` F ) < ( # ` F ) ) )
28		elfzo2 11950	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) e. ( 1..^ ( # ` F ) ) <-> ( ( # ` F ) e. ( ZZ>= ` 1 ) /\ ( # ` F ) e. ZZ /\ ( # ` F ) < ( # ` F ) ) )
29	27, 28	sylnibr 312	. . . . . . . . . . . . . . 15 |- ( F e. Word dom E -> -. ( # ` F ) e. ( 1..^ ( # ` F ) ) )
30		fvex 5889	. . . . . . . . . . . . . . . 16 |- ( # ` F ) e. _V
31		eleq1 2537	. . . . . . . . . . . . . . . . . 18 |- ( x = 0 -> ( x e. ( 1..^ ( # ` F ) ) <-> 0 e. ( 1..^ ( # ` F ) ) ) )
32	31	notbid 301	. . . . . . . . . . . . . . . . 17 |- ( x = 0 -> ( -. x e. ( 1..^ ( # ` F ) ) <-> -. 0 e. ( 1..^ ( # ` F ) ) ) )
33		eleq1 2537	. . . . . . . . . . . . . . . . . 18 |- ( x = ( # ` F ) -> ( x e. ( 1..^ ( # ` F ) ) <-> ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
34	33	notbid 301	. . . . . . . . . . . . . . . . 17 |- ( x = ( # ` F ) -> ( -. x e. ( 1..^ ( # ` F ) ) <-> -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
35	32, 34	ralprg 4012	. . . . . . . . . . . . . . . 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 686	. . . . . . . . . . . . . . 15 |- ( A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) <-> ( -. 0 e. ( 1..^ ( # ` F ) ) /\ -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
37	21, 29, 36	sylanbrc 677	. . . . . . . . . . . . . 14 |- ( F e. Word dom E -> A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) )
38		disj 3809	. . . . . . . . . . . . . 14 |- ( ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) = (/) <-> A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) )
39	37, 38	sylibr 217	. . . . . . . . . . . . 13 |- ( F e. Word dom E -> ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) = (/) )
40	39	imaeq2d 5174	. . . . . . . . . . . 12 |- ( F e. Word dom E -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = ( P " (/) ) )
41		ima0 5189	. . . . . . . . . . . 12 |- ( P " (/) ) = (/)
42	40, 41	syl6eq 2521	. . . . . . . . . . 11 |- ( F e. Word dom E -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = (/) )
43	42	adantr 472	. . . . . . . . . 10 |- ( ( F e. Word dom E /\ Fun `' P ) -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = (/) )
44	13, 43	eqtr3d 2507	. . . . . . . . 9 |- ( ( F e. Word dom E /\ Fun `' P ) -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) )
45	44	ex 441	. . . . . . . 8 |- ( F e. Word dom E -> ( Fun `' P -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) )
46	45	adantr 472	. . . . . . 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 631	. . . . 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 1210	. . . 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 441	. . 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 25378	. . 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 25377	. . 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 276	. 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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   _Vcvv 3031   i^i cin 3389   (/)c0 3722   {cpr 3961   class class class wbr 4395   `'ccnv 4838   dom cdm 4839   |` cres 4841   "cima 4842   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   RRcr 9556   0cc0 9557   1c1 9558   < clt 9693   <_ cle 9694   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810  ..^cfzo 11942   #chash 12553  Word cword 12703   Walks cwalk 25305   Trails ctrail 25306   Paths cpath 25307   SPaths cspath 25308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-wlk 25315  df-trail 25316  df-pth 25317  df-spth 25318

This theorem is referenced by:  spthon  25391  isspthonpth  25393  el2spthonot  25677  usg2wotspth  25691  usgra2pthspth  40173  spthdifv  40174  usgra2pth  40176