Theorem elwspths2spth 26862

Description: A simple path of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 28-Feb-2018.) (Revised by AV, 18-May-2021.)

Hypothesis

Ref	Expression
elwwlks2.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
elwspths2spth	|- ( G e. UPGraph -> ( W e. ( 2 WSPathsN G ) <-> E. a e. V E. b e. V E. c e. V ( W = <" a b c "> /\ E. f E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )

Distinct variable groups:   G, a, b, c, f, p   V, a, b, c, f, p   W, a, b, c, f, p

Proof of Theorem elwspths2spth

Step	Hyp	Ref	Expression
1		2nn0 11309	. . 3 |- 2 e. NN0
2		elwwlks2.v	. . . 4 |- V = (Vtx ` G )
3	2	wspthsnwspthsnon 26811	. . 3 |- ( ( 2 e. NN0 /\ G e. UPGraph ) -> ( W e. ( 2 WSPathsN G ) <-> E. a e. V E. c e. V W e. ( a ( 2 WSPathsNOn G ) c ) ) )
4	1, 3	mpan 706	. 2 |- ( G e. UPGraph -> ( W e. ( 2 WSPathsN G ) <-> E. a e. V E. c e. V W e. ( a ( 2 WSPathsNOn G ) c ) ) )
5	2	elwspths2on 26853	. . . 4 |- ( ( G e. UPGraph /\ a e. V /\ c e. V ) -> ( W e. ( a ( 2 WSPathsNOn G ) c ) <-> E. b e. V ( W = <" a b c "> /\ <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) ) ) )
6	5	3expb 1266	. . 3 |- ( ( G e. UPGraph /\ ( a e. V /\ c e. V ) ) -> ( W e. ( a ( 2 WSPathsNOn G ) c ) <-> E. b e. V ( W = <" a b c "> /\ <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) ) ) )
7	6	2rexbidva 3056	. 2 |- ( G e. UPGraph -> ( E. a e. V E. c e. V W e. ( a ( 2 WSPathsNOn G ) c ) <-> E. a e. V E. c e. V E. b e. V ( W = <" a b c "> /\ <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) ) ) )
8		rexcom 3099	. . . 4 |- ( E. c e. V E. b e. V ( W = <" a b c "> /\ <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) ) <-> E. b e. V E. c e. V ( W = <" a b c "> /\ <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) ) )
9		simpr 477	. . . . . . . . . 10 |- ( ( G e. UPGraph /\ a e. V ) -> a e. V )
10		simpr 477	. . . . . . . . . 10 |- ( ( b e. V /\ c e. V ) -> c e. V )
11	9, 10	anim12i 590	. . . . . . . . 9 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( a e. V /\ c e. V ) )
12	2	wspthnon 26743	. . . . . . . . 9 |- ( ( a e. V /\ c e. V ) -> ( <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) <-> ( <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) /\ E. f f ( a (SPathsOn ` G ) c ) <" a b c "> ) ) )
13	11, 12	syl 17	. . . . . . . 8 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) <-> ( <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) /\ E. f f ( a (SPathsOn ` G ) c ) <" a b c "> ) ) )
14	13	adantr 481	. . . . . . 7 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) <-> ( <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) /\ E. f f ( a (SPathsOn ` G ) c ) <" a b c "> ) ) )
15		ancom 466	. . . . . . . . 9 |- ( ( <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) /\ E. f f ( a (SPathsOn ` G ) c ) <" a b c "> ) <-> ( E. f f ( a (SPathsOn ` G ) c ) <" a b c "> /\ <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) ) )
16		19.41v 1914	. . . . . . . . 9 |- ( E. f ( f ( a (SPathsOn ` G ) c ) <" a b c "> /\ <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) ) <-> ( E. f f ( a (SPathsOn ` G ) c ) <" a b c "> /\ <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) ) )
17	15, 16	bitr4i 267	. . . . . . . 8 |- ( ( <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) /\ E. f f ( a (SPathsOn ` G ) c ) <" a b c "> ) <-> E. f ( f ( a (SPathsOn ` G ) c ) <" a b c "> /\ <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) ) )
18		vex 3203	. . . . . . . . . . . . . 14 |- f e. _V
19		s3cli 13626	. . . . . . . . . . . . . 14 |- <" a b c "> e. Word _V
20	18, 19	pm3.2i 471	. . . . . . . . . . . . 13 |- ( f e. _V /\ <" a b c "> e. Word _V )
21	2	isspthonpth 26645	. . . . . . . . . . . . 13 |- ( ( ( a e. V /\ c e. V ) /\ ( f e. _V /\ <" a b c "> e. Word _V ) ) -> ( f ( a (SPathsOn ` G ) c ) <" a b c "> <-> ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) ) )
22	11, 20, 21	sylancl 694	. . . . . . . . . . . 12 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( f ( a (SPathsOn ` G ) c ) <" a b c "> <-> ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) ) )
23	2	wwlknon 26742	. . . . . . . . . . . . . 14 |- ( ( a e. V /\ c e. V ) -> ( <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) <-> ( <" a b c "> e. ( 2 WWalksN G ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) )
24	11, 23	syl 17	. . . . . . . . . . . . 13 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) <-> ( <" a b c "> e. ( 2 WWalksN G ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) )
25		iswwlksn 26730	. . . . . . . . . . . . . . . 16 |- ( 2 e. NN0 -> ( <" a b c "> e. ( 2 WWalksN G ) <-> ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) ) )
26	1, 25	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( <" a b c "> e. ( 2 WWalksN G ) <-> ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) )
27	26	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( <" a b c "> e. ( 2 WWalksN G ) <-> ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) ) )
28	27	3anbi1d 1403	. . . . . . . . . . . . 13 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( ( <" a b c "> e. ( 2 WWalksN G ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) <-> ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) )
29	24, 28	bitrd 268	. . . . . . . . . . . 12 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) <-> ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) )
30	22, 29	anbi12d 747	. . . . . . . . . . 11 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( ( f ( a (SPathsOn ` G ) c ) <" a b c "> /\ <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) ) <-> ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) ) )
31	30	adantr 481	. . . . . . . . . 10 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( ( f ( a (SPathsOn ` G ) c ) <" a b c "> /\ <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) ) <-> ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) ) )
32	19	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> <" a b c "> e. Word _V )
33		simprl1 1106	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ p = <" a b c "> ) /\ ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) ) -> f (SPaths ` G ) <" a b c "> )
34		spthiswlk 26624	. . . . . . . . . . . . . . . . . . . 20 |- ( f (SPaths ` G ) <" a b c "> -> f (Walks ` G ) <" a b c "> )
35		wlklenvm1 26517	. . . . . . . . . . . . . . . . . . . 20 |- ( f (Walks ` G ) <" a b c "> -> ( # ` f ) = ( ( # ` <" a b c "> ) - 1 ) )
36		simpl 473	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( # ` f ) = ( ( # ` <" a b c "> ) - 1 ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) -> ( # ` f ) = ( ( # ` <" a b c "> ) - 1 ) )
37		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( # ` <" a b c "> ) = ( 2 + 1 ) -> ( ( # ` <" a b c "> ) - 1 ) = ( ( 2 + 1 ) - 1 ) )
38		2cn 11091	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- 2 e. CC
39		pncan1 10454	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( 2 e. CC -> ( ( 2 + 1 ) - 1 ) = 2 )
40	38, 39	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( 2 + 1 ) - 1 ) = 2
41	37, 40	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` <" a b c "> ) = ( 2 + 1 ) -> ( ( # ` <" a b c "> ) - 1 ) = 2 )
42	41	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) -> ( ( # ` <" a b c "> ) - 1 ) = 2 )
43	42	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) -> ( ( # ` <" a b c "> ) - 1 ) = 2 )
44	43	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( # ` f ) = ( ( # ` <" a b c "> ) - 1 ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) -> ( ( # ` <" a b c "> ) - 1 ) = 2 )
45	36, 44	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( # ` f ) = ( ( # ` <" a b c "> ) - 1 ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) -> ( # ` f ) = 2 )
46	45	ex 450	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` f ) = ( ( # ` <" a b c "> ) - 1 ) -> ( ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) -> ( # ` f ) = 2 ) )
47	34, 35, 46	3syl 18	. . . . . . . . . . . . . . . . . . 19 |- ( f (SPaths ` G ) <" a b c "> -> ( ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) -> ( # ` f ) = 2 ) )
48	47	3ad2ant1 1082	. . . . . . . . . . . . . . . . . 18 |- ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) -> ( ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) -> ( # ` f ) = 2 ) )
49	48	imp 445	. . . . . . . . . . . . . . . . 17 |- ( ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) -> ( # ` f ) = 2 )
50	49	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ p = <" a b c "> ) /\ ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) ) -> ( # ` f ) = 2 )
51		vex 3203	. . . . . . . . . . . . . . . . . . . 20 |- a e. _V
52		s3fv0 13636	. . . . . . . . . . . . . . . . . . . 20 |- ( a e. _V -> ( <" a b c "> ` 0 ) = a )
53	51, 52	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- ( <" a b c "> ` 0 ) = a
54	53	eqcomi 2631	. . . . . . . . . . . . . . . . . 18 |- a = ( <" a b c "> ` 0 )
55		vex 3203	. . . . . . . . . . . . . . . . . . . 20 |- b e. _V
56		s3fv1 13637	. . . . . . . . . . . . . . . . . . . 20 |- ( b e. _V -> ( <" a b c "> ` 1 ) = b )
57	55, 56	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- ( <" a b c "> ` 1 ) = b
58	57	eqcomi 2631	. . . . . . . . . . . . . . . . . 18 |- b = ( <" a b c "> ` 1 )
59		vex 3203	. . . . . . . . . . . . . . . . . . . 20 |- c e. _V
60		s3fv2 13638	. . . . . . . . . . . . . . . . . . . 20 |- ( c e. _V -> ( <" a b c "> ` 2 ) = c )
61	59, 60	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- ( <" a b c "> ` 2 ) = c
62	61	eqcomi 2631	. . . . . . . . . . . . . . . . . 18 |- c = ( <" a b c "> ` 2 )
63	54, 58, 62	3pm3.2i 1239	. . . . . . . . . . . . . . . . 17 |- ( a = ( <" a b c "> ` 0 ) /\ b = ( <" a b c "> ` 1 ) /\ c = ( <" a b c "> ` 2 ) )
64	63	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ p = <" a b c "> ) /\ ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) ) -> ( a = ( <" a b c "> ` 0 ) /\ b = ( <" a b c "> ` 1 ) /\ c = ( <" a b c "> ` 2 ) ) )
65	33, 50, 64	3jca 1242	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ p = <" a b c "> ) /\ ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) ) -> ( f (SPaths ` G ) <" a b c "> /\ ( # ` f ) = 2 /\ ( a = ( <" a b c "> ` 0 ) /\ b = ( <" a b c "> ` 1 ) /\ c = ( <" a b c "> ` 2 ) ) ) )
66		breq2 4657	. . . . . . . . . . . . . . . . 17 |- ( p = <" a b c "> -> ( f (SPaths ` G ) p <-> f (SPaths ` G ) <" a b c "> ) )
67		fveq1 6190	. . . . . . . . . . . . . . . . . . 19 |- ( p = <" a b c "> -> ( p ` 0 ) = ( <" a b c "> ` 0 ) )
68	67	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 |- ( p = <" a b c "> -> ( a = ( p ` 0 ) <-> a = ( <" a b c "> ` 0 ) ) )
69		fveq1 6190	. . . . . . . . . . . . . . . . . . 19 |- ( p = <" a b c "> -> ( p ` 1 ) = ( <" a b c "> ` 1 ) )
70	69	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 |- ( p = <" a b c "> -> ( b = ( p ` 1 ) <-> b = ( <" a b c "> ` 1 ) ) )
71		fveq1 6190	. . . . . . . . . . . . . . . . . . 19 |- ( p = <" a b c "> -> ( p ` 2 ) = ( <" a b c "> ` 2 ) )
72	71	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 |- ( p = <" a b c "> -> ( c = ( p ` 2 ) <-> c = ( <" a b c "> ` 2 ) ) )
73	68, 70, 72	3anbi123d 1399	. . . . . . . . . . . . . . . . 17 |- ( p = <" a b c "> -> ( ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) <-> ( a = ( <" a b c "> ` 0 ) /\ b = ( <" a b c "> ` 1 ) /\ c = ( <" a b c "> ` 2 ) ) ) )
74	66, 73	3anbi13d 1401	. . . . . . . . . . . . . . . 16 |- ( p = <" a b c "> -> ( ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) <-> ( f (SPaths ` G ) <" a b c "> /\ ( # ` f ) = 2 /\ ( a = ( <" a b c "> ` 0 ) /\ b = ( <" a b c "> ` 1 ) /\ c = ( <" a b c "> ` 2 ) ) ) ) )
75	74	ad2antlr 763	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ p = <" a b c "> ) /\ ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) ) -> ( ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) <-> ( f (SPaths ` G ) <" a b c "> /\ ( # ` f ) = 2 /\ ( a = ( <" a b c "> ` 0 ) /\ b = ( <" a b c "> ` 1 ) /\ c = ( <" a b c "> ` 2 ) ) ) ) )
76	65, 75	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ p = <" a b c "> ) /\ ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) ) -> ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) )
77	76	ex 450	. . . . . . . . . . . . 13 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ p = <" a b c "> ) -> ( ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) -> ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
78	32, 77	spcimedv 3292	. . . . . . . . . . . 12 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) -> E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
79		spthiswlk 26624	. . . . . . . . . . . . . . . . . . . . 21 |- ( f (SPaths ` G ) p -> f (Walks ` G ) p )
80		wlklenvp1 26514	. . . . . . . . . . . . . . . . . . . . 21 |- ( f (Walks ` G ) p -> ( # ` p ) = ( ( # ` f ) + 1 ) )
81		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( # ` f ) = 2 -> ( ( # ` f ) + 1 ) = ( 2 + 1 ) )
82		2p1e3 11151	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 2 + 1 ) = 3
83	81, 82	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( # ` f ) = 2 -> ( ( # ` f ) + 1 ) = 3 )
84	83	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` f ) = 2 -> ( ( # ` p ) = ( ( # ` f ) + 1 ) <-> ( # ` p ) = 3 ) )
85	84	biimpcd 239	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` p ) = ( ( # ` f ) + 1 ) -> ( ( # ` f ) = 2 -> ( # ` p ) = 3 ) )
86	79, 80, 85	3syl 18	. . . . . . . . . . . . . . . . . . . 20 |- ( f (SPaths ` G ) p -> ( ( # ` f ) = 2 -> ( # ` p ) = 3 ) )
87	86	imp 445	. . . . . . . . . . . . . . . . . . 19 |- ( ( f (SPaths ` G ) p /\ ( # ` f ) = 2 ) -> ( # ` p ) = 3 )
88	87	3adant3 1081	. . . . . . . . . . . . . . . . . 18 |- ( ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( # ` p ) = 3 )
89	88	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) -> ( # ` p ) = 3 )
90		eqcom 2629	. . . . . . . . . . . . . . . . . . . . 21 |- ( a = ( p ` 0 ) <-> ( p ` 0 ) = a )
91		eqcom 2629	. . . . . . . . . . . . . . . . . . . . 21 |- ( b = ( p ` 1 ) <-> ( p ` 1 ) = b )
92		eqcom 2629	. . . . . . . . . . . . . . . . . . . . 21 |- ( c = ( p ` 2 ) <-> ( p ` 2 ) = c )
93	90, 91, 92	3anbi123i 1251	. . . . . . . . . . . . . . . . . . . 20 |- ( ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) <-> ( ( p ` 0 ) = a /\ ( p ` 1 ) = b /\ ( p ` 2 ) = c ) )
94	93	biimpi 206	. . . . . . . . . . . . . . . . . . 19 |- ( ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) -> ( ( p ` 0 ) = a /\ ( p ` 1 ) = b /\ ( p ` 2 ) = c ) )
95	94	3ad2ant3 1084	. . . . . . . . . . . . . . . . . 18 |- ( ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( ( p ` 0 ) = a /\ ( p ` 1 ) = b /\ ( p ` 2 ) = c ) )
96	95	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) -> ( ( p ` 0 ) = a /\ ( p ` 1 ) = b /\ ( p ` 2 ) = c ) )
97	89, 96	jca 554	. . . . . . . . . . . . . . . 16 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) -> ( ( # ` p ) = 3 /\ ( ( p ` 0 ) = a /\ ( p ` 1 ) = b /\ ( p ` 2 ) = c ) ) )
98	2	wlkpwrd 26513	. . . . . . . . . . . . . . . . . . 19 |- ( f (Walks ` G ) p -> p e. Word V )
99	79, 98	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( f (SPaths ` G ) p -> p e. Word V )
100	99	3ad2ant1 1082	. . . . . . . . . . . . . . . . 17 |- ( ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> p e. Word V )
101	9	anim1i 592	. . . . . . . . . . . . . . . . . 18 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( a e. V /\ ( b e. V /\ c e. V ) ) )
102		3anass 1042	. . . . . . . . . . . . . . . . . 18 |- ( ( a e. V /\ b e. V /\ c e. V ) <-> ( a e. V /\ ( b e. V /\ c e. V ) ) )
103	101, 102	sylibr 224	. . . . . . . . . . . . . . . . 17 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( a e. V /\ b e. V /\ c e. V ) )
104		eqwrds3 13704	. . . . . . . . . . . . . . . . 17 |- ( ( p e. Word V /\ ( a e. V /\ b e. V /\ c e. V ) ) -> ( p = <" a b c "> <-> ( ( # ` p ) = 3 /\ ( ( p ` 0 ) = a /\ ( p ` 1 ) = b /\ ( p ` 2 ) = c ) ) ) )
105	100, 103, 104	syl2anr 495	. . . . . . . . . . . . . . . 16 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) -> ( p = <" a b c "> <-> ( ( # ` p ) = 3 /\ ( ( p ` 0 ) = a /\ ( p ` 1 ) = b /\ ( p ` 2 ) = c ) ) ) )
106	97, 105	mpbird 247	. . . . . . . . . . . . . . 15 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) -> p = <" a b c "> )
107	66	biimpcd 239	. . . . . . . . . . . . . . . . . . . 20 |- ( f (SPaths ` G ) p -> ( p = <" a b c "> -> f (SPaths ` G ) <" a b c "> ) )
108	107	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . 19 |- ( ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( p = <" a b c "> -> f (SPaths ` G ) <" a b c "> ) )
109	108	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) -> ( p = <" a b c "> -> f (SPaths ` G ) <" a b c "> ) )
110	109	imp 445	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) /\ p = <" a b c "> ) -> f (SPaths ` G ) <" a b c "> )
111	53	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) /\ p = <" a b c "> ) -> ( <" a b c "> ` 0 ) = a )
112		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` f ) = 2 -> ( <" a b c "> ` ( # ` f ) ) = ( <" a b c "> ` 2 ) )
113	112, 61	syl6eq 2672	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` f ) = 2 -> ( <" a b c "> ` ( # ` f ) ) = c )
114	113	3ad2ant2 1083	. . . . . . . . . . . . . . . . . 18 |- ( ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( <" a b c "> ` ( # ` f ) ) = c )
115	114	ad2antlr 763	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) /\ p = <" a b c "> ) -> ( <" a b c "> ` ( # ` f ) ) = c )
116	110, 111, 115	3jca 1242	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) /\ p = <" a b c "> ) -> ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) )
117		wlkiswwlks1 26753	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( G e. UPGraph -> ( f (Walks ` G ) p -> p e. (WWalks ` G ) ) )
118	117	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( G e. UPGraph /\ a e. V ) -> ( f (Walks ` G ) p -> p e. (WWalks ` G ) ) )
119	118	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( f (Walks ` G ) p -> p e. (WWalks ` G ) ) )
120	79, 119	syl5com 31	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f (SPaths ` G ) p -> ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> p e. (WWalks ` G ) ) )
121	120	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> p e. (WWalks ` G ) ) )
122	121	impcom 446	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) -> p e. (WWalks ` G ) )
123	122	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) /\ p = <" a b c "> ) -> p e. (WWalks ` G ) )
124		eleq1 2689	. . . . . . . . . . . . . . . . . . . . 21 |- ( p = <" a b c "> -> ( p e. (WWalks ` G ) <-> <" a b c "> e. (WWalks ` G ) ) )
125	124	bicomd 213	. . . . . . . . . . . . . . . . . . . 20 |- ( p = <" a b c "> -> ( <" a b c "> e. (WWalks ` G ) <-> p e. (WWalks ` G ) ) )
126	125	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) /\ p = <" a b c "> ) -> ( <" a b c "> e. (WWalks ` G ) <-> p e. (WWalks ` G ) ) )
127	123, 126	mpbird 247	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) /\ p = <" a b c "> ) -> <" a b c "> e. (WWalks ` G ) )
128		s3len 13639	. . . . . . . . . . . . . . . . . . 19 |- ( # ` <" a b c "> ) = 3
129		df-3 11080	. . . . . . . . . . . . . . . . . . 19 |- 3 = ( 2 + 1 )
130	128, 129	eqtri 2644	. . . . . . . . . . . . . . . . . 18 |- ( # ` <" a b c "> ) = ( 2 + 1 )
131	127, 130	jctir 561	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) /\ p = <" a b c "> ) -> ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) )
132	61	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) /\ p = <" a b c "> ) -> ( <" a b c "> ` 2 ) = c )
133	131, 111, 132	3jca 1242	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) /\ p = <" a b c "> ) -> ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) )
134	116, 133	jca 554	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) /\ p = <" a b c "> ) -> ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) )
135	106, 134	mpdan 702	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) -> ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) )
136	135	ex 450	. . . . . . . . . . . . 13 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) ) )
137	136	exlimdv 1861	. . . . . . . . . . . 12 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) ) )
138	78, 137	impbid 202	. . . . . . . . . . 11 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) <-> E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
139	138	adantr 481	. . . . . . . . . 10 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( ( ( f (SPaths ` G ) <" a b c "> /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` ( # ` f ) ) = c ) /\ ( ( <" a b c "> e. (WWalks ` G ) /\ ( # ` <" a b c "> ) = ( 2 + 1 ) ) /\ ( <" a b c "> ` 0 ) = a /\ ( <" a b c "> ` 2 ) = c ) ) <-> E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
140	31, 139	bitrd 268	. . . . . . . . 9 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( ( f ( a (SPathsOn ` G ) c ) <" a b c "> /\ <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) ) <-> E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
141	140	exbidv 1850	. . . . . . . 8 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( E. f ( f ( a (SPathsOn ` G ) c ) <" a b c "> /\ <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) ) <-> E. f E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
142	17, 141	syl5bb 272	. . . . . . 7 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( ( <" a b c "> e. ( a ( 2 WWalksNOn G ) c ) /\ E. f f ( a (SPathsOn ` G ) c ) <" a b c "> ) <-> E. f E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
143	14, 142	bitrd 268	. . . . . 6 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) <-> E. f E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
144	143	pm5.32da 673	. . . . 5 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( ( W = <" a b c "> /\ <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) ) <-> ( W = <" a b c "> /\ E. f E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )
145	144	2rexbidva 3056	. . . 4 |- ( ( G e. UPGraph /\ a e. V ) -> ( E. b e. V E. c e. V ( W = <" a b c "> /\ <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) ) <-> E. b e. V E. c e. V ( W = <" a b c "> /\ E. f E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )
146	8, 145	syl5bb 272	. . 3 |- ( ( G e. UPGraph /\ a e. V ) -> ( E. c e. V E. b e. V ( W = <" a b c "> /\ <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) ) <-> E. b e. V E. c e. V ( W = <" a b c "> /\ E. f E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )
147	146	rexbidva 3049	. 2 |- ( G e. UPGraph -> ( E. a e. V E. c e. V E. b e. V ( W = <" a b c "> /\ <" a b c "> e. ( a ( 2 WSPathsNOn G ) c ) ) <-> E. a e. V E. b e. V E. c e. V ( W = <" a b c "> /\ E. f E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )
148	4, 7, 147	3bitrd 294	1 |- ( G e. UPGraph -> ( W e. ( 2 WSPathsN G ) <-> E. a e. V E. b e. V E. c e. V ( W = <" a b c "> /\ E. f E. p ( f (SPaths ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   2c2 11070   3c3 11071   NN0cn0 11292   #chash 13117  Word cword 13291   <"cs3 13587  Vtxcvtx 25874   UPGraph cupgr 25975  Walkscwlks 26492  SPathscspths 26609  SPathsOncspthson 26611  WWalkscwwlks 26717   WWalksN cwwlksn 26718   WWalksNOn cwwlksnon 26719   WSPathsN cwwspthsn 26720   WSPathsNOn cwwspthsnon 26721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-spthson 26615  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-wspthsn 26725  df-wspthsnon 26726

This theorem is referenced by: (None)