Theorem spthispth 24239

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 24175	. . 3 |- SPaths = ( v e. _V , e e. _V |-> { <. f , p >. | ( f ( v Trails e ) p /\ Fun `' p ) } )
2	1	brovmpt2ex 6943	. 2 |- ( F ( V SPaths E ) P -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )
3		simprl 755	. . . . 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 5649	. . . . . . 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 24205	. . . . . . . . . 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 24185	. . . . . . . . 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 5657	. . . . . . . . . . 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 10066	. . . . . . . . . . . . . . . . . 18 |- 0 < 1
15		0re 9587	. . . . . . . . . . . . . . . . . . 19 |- 0 e. RR
16		1re 9586	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
17	15, 16	ltnlei 9696	. . . . . . . . . . . . . . . . . 18 |- ( 0 < 1 <-> -. 1 <_ 0 )
18	14, 17	mpbi 208	. . . . . . . . . . . . . . . . 17 |- -. 1 <_ 0
19		elfzole1 11795	. . . . . . . . . . . . . . . . 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 12516	. . . . . . . . . . . . . . . . . . . 20 |- ( F e. Word dom E -> F e. Fin )
23		hashcl 12385	. . . . . . . . . . . . . . . . . . . . 21 |- ( F e. Fin -> ( # ` F ) e. NN0 )
24	23	nn0red 10844	. . . . . . . . . . . . . . . . . . . 20 |- ( F e. Fin -> ( # ` F ) e. RR )
25	22, 24	syl 16	. . . . . . . . . . . . . . . . . . 19 |- ( F e. Word dom E -> ( # ` F ) e. RR )
26	25	ltnrd 9709	. . . . . . . . . . . . . . . . . 18 |- ( F e. Word dom E -> -. ( # ` F ) < ( # ` F ) )
27		3mix3 1162	. . . . . . . . . . . . . . . . . 18 |- ( -. ( # ` F ) < ( # ` F ) -> ( -. ( # ` F ) e. ( ZZ>= ` 1 ) \/ -. ( # ` F ) e. ZZ \/ -. ( # ` F ) < ( # ` F ) ) )
28	26, 27	syl 16	. . . . . . . . . . . . . . . . 17 |- ( F e. Word dom E -> ( -. ( # ` F ) e. ( ZZ>= ` 1 ) \/ -. ( # ` F ) e. ZZ \/ -. ( # ` F ) < ( # ` F ) ) )
29		3ianor 985	. . . . . . . . . . . . . . . . 17 |- ( -. ( ( # ` F ) e. ( ZZ>= ` 1 ) /\ ( # ` F ) e. ZZ /\ ( # ` F ) < ( # ` F ) ) <-> ( -. ( # ` F ) e. ( ZZ>= ` 1 ) \/ -. ( # ` F ) e. ZZ \/ -. ( # ` F ) < ( # ` F ) ) )
30	28, 29	sylibr 212	. . . . . . . . . . . . . . . 16 |- ( F e. Word dom E -> -. ( ( # ` F ) e. ( ZZ>= ` 1 ) /\ ( # ` F ) e. ZZ /\ ( # ` F ) < ( # ` F ) ) )
31		elfzo2 11791	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) e. ( 1..^ ( # ` F ) ) <-> ( ( # ` F ) e. ( ZZ>= ` 1 ) /\ ( # ` F ) e. ZZ /\ ( # ` F ) < ( # ` F ) ) )
32	30, 31	sylnibr 305	. . . . . . . . . . . . . . 15 |- ( F e. Word dom E -> -. ( # ` F ) e. ( 1..^ ( # ` F ) ) )
33		fvex 5869	. . . . . . . . . . . . . . . 16 |- ( # ` F ) e. _V
34		eleq1 2534	. . . . . . . . . . . . . . . . . 18 |- ( x = 0 -> ( x e. ( 1..^ ( # ` F ) ) <-> 0 e. ( 1..^ ( # ` F ) ) ) )
35	34	notbid 294	. . . . . . . . . . . . . . . . 17 |- ( x = 0 -> ( -. x e. ( 1..^ ( # ` F ) ) <-> -. 0 e. ( 1..^ ( # ` F ) ) ) )
36		eleq1 2534	. . . . . . . . . . . . . . . . . 18 |- ( x = ( # ` F ) -> ( x e. ( 1..^ ( # ` F ) ) <-> ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
37	36	notbid 294	. . . . . . . . . . . . . . . . 17 |- ( x = ( # ` F ) -> ( -. x e. ( 1..^ ( # ` F ) ) <-> -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
38	35, 37	ralprg 4071	. . . . . . . . . . . . . . . 16 |- ( ( 0 e. RR /\ ( # ` F ) e. _V ) -> ( A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) <-> ( -. 0 e. ( 1..^ ( # ` F ) ) /\ -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) ) )
39	15, 33, 38	mp2an 672	. . . . . . . . . . . . . . 15 |- ( A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) <-> ( -. 0 e. ( 1..^ ( # ` F ) ) /\ -. ( # ` F ) e. ( 1..^ ( # ` F ) ) ) )
40	21, 32, 39	sylanbrc 664	. . . . . . . . . . . . . 14 |- ( F e. Word dom E -> A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) )
41		disj 3862	. . . . . . . . . . . . . 14 |- ( ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) = (/) <-> A. x e. { 0 , ( # ` F ) } -. x e. ( 1..^ ( # ` F ) ) )
42	40, 41	sylibr 212	. . . . . . . . . . . . 13 |- ( F e. Word dom E -> ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) = (/) )
43	42	imaeq2d 5330	. . . . . . . . . . . 12 |- ( F e. Word dom E -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = ( P " (/) ) )
44		ima0 5345	. . . . . . . . . . . 12 |- ( P " (/) ) = (/)
45	43, 44	syl6eq 2519	. . . . . . . . . . 11 |- ( F e. Word dom E -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = (/) )
46	45	adantr 465	. . . . . . . . . 10 |- ( ( F e. Word dom E /\ Fun `' P ) -> ( P " ( { 0 , ( # ` F ) } i^i ( 1..^ ( # ` F ) ) ) ) = (/) )
47	13, 46	eqtr3d 2505	. . . . . . . . 9 |- ( ( F e. Word dom E /\ Fun `' P ) -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) )
48	47	ex 434	. . . . . . . 8 |- ( F e. Word dom E -> ( Fun `' P -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) )
49	48	adantr 465	. . . . . . 7 |- ( ( F e. Word dom E /\ P : ( 0 ... ( # ` F ) ) --> V ) -> ( Fun `' P -> ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1..^ ( # ` F ) ) ) ) = (/) ) )
50	11, 49	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 ) ) ) ) = (/) ) )
51	50	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 ) ) ) ) = (/) )
52	3, 6, 51	3jca 1171	. . . 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 ) ) ) ) = (/) ) )
53	52	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 ) ) ) ) = (/) ) ) )
54		isspth 24235	. . 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 ) ) )
55		ispth 24234	. . 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 ) ) ) ) = (/) ) ) )
56	53, 54, 55	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 ) )
57	2, 56	mpcom 36	1 |- ( F ( V SPaths E ) P -> F ( V Paths E ) P )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   \/ w3o 967   /\ w3a 968   = wceq 1374   e. wcel 1762   A.wral 2809   _Vcvv 3108   i^i cin 3470   (/)c0 3780   {cpr 4024   class class class wbr 4442   `'ccnv 4993   dom cdm 4994   |` cres 4996   "cima 4997   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6277   Fincfn 7508   RRcr 9482   0cc0 9483   1c1 9484   < clt 9619   <_ cle 9620   ZZcz 10855   ZZ>=cuz 11073   ...cfz 11663  ..^cfzo 11783   #chash 12362  Word cword 12489   Walks cwalk 24162   Trails ctrail 24163   Paths cpath 24164   SPaths cspath 24165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-fzo 11784  df-hash 12363  df-word 12497  df-wlk 24172  df-trail 24173  df-pth 24174  df-spth 24175

This theorem is referenced by:  spthon  24248  isspthonpth  24250  el2spthonot  24534  usg2wotspth  24548  usgra2pthspth  31777  spthdifv  31778  usgra2pth  31780