Theorem usgra2wlkspthlem2 30444

Description: Lemma 2 for usgra2wlkspth 30445. (Contributed by Alexander van der Vekens, 2-Mar-2018.)

Assertion

Ref	Expression
usgra2wlkspthlem2	|- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ ( V USGrph E /\ P : ( 0 ... ( # ` F ) ) --> V ) ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' P ) )

Distinct variable groups:   i, E   i, F   P, i

Allowed substitution hints:   A( i)   B( i)   V( i)

Proof of Theorem usgra2wlkspthlem2

Step	Hyp	Ref	Expression
1		oveq2 6207	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 2 -> ( 0 ... ( # ` F ) ) = ( 0 ... 2 ) )
2	1	feq2d 5654	. . . . . . . . . . . . . 14 |- ( ( # ` F ) = 2 -> ( P : ( 0 ... ( # ` F ) ) --> V <-> P : ( 0 ... 2 ) --> V ) )
3		fz0tp 11629	. . . . . . . . . . . . . . . 16 |- ( 0 ... 2 ) = { 0 , 1 , 2 }
4	3	feq2i 5659	. . . . . . . . . . . . . . 15 |- ( P : ( 0 ... 2 ) --> V <-> P : { 0 , 1 , 2 } --> V )
5	4	biimpi 194	. . . . . . . . . . . . . 14 |- ( P : ( 0 ... 2 ) --> V -> P : { 0 , 1 , 2 } --> V )
6	2, 5	syl6bi 228	. . . . . . . . . . . . 13 |- ( ( # ` F ) = 2 -> ( P : ( 0 ... ( # ` F ) ) --> V -> P : { 0 , 1 , 2 } --> V ) )
7	6	ad2antll 728	. . . . . . . . . . . 12 |- ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> P : { 0 , 1 , 2 } --> V ) )
8	7	adantr 465	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> P : { 0 , 1 , 2 } --> V ) )
9	8	impcom 430	. . . . . . . . . 10 |- ( ( P : ( 0 ... ( # ` F ) ) --> V /\ ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) -> P : { 0 , 1 , 2 } --> V )
10		eqeq12 2473	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( P ` 0 ) = ( P ` 2 ) <-> A = B ) )
11	10	biimpd 207	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( P ` 0 ) = ( P ` 2 ) -> A = B ) )
12	11	necon3d 2675	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( A =/= B -> ( P ` 0 ) =/= ( P ` 2 ) ) )
13	12	3impia 1185	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) -> ( P ` 0 ) =/= ( P ` 2 ) )
14	13	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P ` 0 ) =/= ( P ` 2 ) )
15	14	adantl 466	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> ( P ` 0 ) =/= ( P ` 2 ) )
16	15	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) /\ P : ( 0 ... 2 ) --> V ) -> ( P ` 0 ) =/= ( P ` 2 ) )
17		simplr 754	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) -> V USGrph E )
18		usgrafun 23428	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( V USGrph E -> Fun E )
19		wrdf 12357	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( F e. Word dom E -> F : ( 0..^ ( # ` F ) ) --> dom E )
20		oveq2 6207	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = ( 0..^ 2 ) )
21	20	feq2d 5654	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E <-> F : ( 0..^ 2 ) --> dom E ) )
22		c0ex 9490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- 0 e. _V
23	22	prid1 4090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 0 e. { 0 , 1 }
24		fzo0to2pr 11730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( 0..^ 2 ) = { 0 , 1 }
25	23, 24	eleqtrri 2541	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- 0 e. ( 0..^ 2 )
26		ffvelrn 5949	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( F : ( 0..^ 2 ) --> dom E /\ 0 e. ( 0..^ 2 ) ) -> ( F ` 0 ) e. dom E )
27	25, 26	mpan2 671	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( F : ( 0..^ 2 ) --> dom E -> ( F ` 0 ) e. dom E )
28	21, 27	syl6bi 228	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( F ` 0 ) e. dom E ) )
29	19, 28	mpan9 469	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( F e. Word dom E /\ ( # ` F ) = 2 ) -> ( F ` 0 ) e. dom E )
30		fvelrn 5947	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( Fun E /\ ( F ` 0 ) e. dom E ) -> ( E ` ( F ` 0 ) ) e. ran E )
31	18, 29, 30	syl2anr 478	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( E ` ( F ` 0 ) ) e. ran E )
32	31	adantr 465	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) -> ( E ` ( F ` 0 ) ) e. ran E )
33		eleq1 2526	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } -> ( ( E ` ( F ` 0 ) ) e. ran E <-> { ( P ` 0 ) , ( P ` 1 ) } e. ran E ) )
34	33	adantl 466	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) -> ( ( E ` ( F ` 0 ) ) e. ran E <-> { ( P ` 0 ) , ( P ` 1 ) } e. ran E ) )
35	32, 34	mpbid 210	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) -> { ( P ` 0 ) , ( P ` 1 ) } e. ran E )
36		usgraedgrn 23451	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( V USGrph E /\ { ( P ` 0 ) , ( P ` 1 ) } e. ran E ) -> ( P ` 0 ) =/= ( P ` 1 ) )
37	17, 35, 36	syl2anc 661	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) -> ( P ` 0 ) =/= ( P ` 1 ) )
38	37	ex 434	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } -> ( P ` 0 ) =/= ( P ` 1 ) ) )
39		simplr 754	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> V USGrph E )
40		1ex 9491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- 1 e. _V
41	40	prid2 4091	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 1 e. { 0 , 1 }
42	41, 24	eleqtrri 2541	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- 1 e. ( 0..^ 2 )
43		ffvelrn 5949	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( F : ( 0..^ 2 ) --> dom E /\ 1 e. ( 0..^ 2 ) ) -> ( F ` 1 ) e. dom E )
44	42, 43	mpan2 671	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( F : ( 0..^ 2 ) --> dom E -> ( F ` 1 ) e. dom E )
45	21, 44	syl6bi 228	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( F ` 1 ) e. dom E ) )
46	19, 45	mpan9 469	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( F e. Word dom E /\ ( # ` F ) = 2 ) -> ( F ` 1 ) e. dom E )
47		fvelrn 5947	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( Fun E /\ ( F ` 1 ) e. dom E ) -> ( E ` ( F ` 1 ) ) e. ran E )
48	18, 46, 47	syl2anr 478	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( E ` ( F ` 1 ) ) e. ran E )
49	48	adantr 465	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( E ` ( F ` 1 ) ) e. ran E )
50		eleq1 2526	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } -> ( ( E ` ( F ` 1 ) ) e. ran E <-> { ( P ` 1 ) , ( P ` 2 ) } e. ran E ) )
51	50	adantl 466	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( E ` ( F ` 1 ) ) e. ran E <-> { ( P ` 1 ) , ( P ` 2 ) } e. ran E ) )
52	49, 51	mpbid 210	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> { ( P ` 1 ) , ( P ` 2 ) } e. ran E )
53		usgraedgrn 23451	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( V USGrph E /\ { ( P ` 1 ) , ( P ` 2 ) } e. ran E ) -> ( P ` 1 ) =/= ( P ` 2 ) )
54	39, 52, 53	syl2anc 661	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 1 ) =/= ( P ` 2 ) )
55	54	ex 434	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } -> ( P ` 1 ) =/= ( P ` 2 ) ) )
56	38, 55	anim12d 563	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) )
57	56	adantld 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) )
58	57	imp 429	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) )
59	58	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) /\ P : ( 0 ... 2 ) --> V ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) )
60		3anan12 978	. . . . . . . . . . . . . . . 16 |- ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) <-> ( ( P ` 0 ) =/= ( P ` 2 ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) )
61	16, 59, 60	sylanbrc 664	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) /\ P : ( 0 ... 2 ) --> V ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) )
62	61	exp41 610	. . . . . . . . . . . . . 14 |- ( ( F e. Word dom E /\ ( # ` F ) = 2 ) -> ( V USGrph E -> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P : ( 0 ... 2 ) --> V -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) ) ) )
63	62	impcom 430	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P : ( 0 ... 2 ) --> V -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) ) )
64		fveq2 5798	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = ( P ` 2 ) )
65	64	eqeq1d 2456	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 2 -> ( ( P ` ( # ` F ) ) = B <-> ( P ` 2 ) = B ) )
66	65	3anbi2d 1295	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) <-> ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) ) )
67	20, 24	syl6eq 2511	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = { 0 , 1 } )
68	67	raleqdv 3027	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 2 -> ( A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> A. i e. { 0 , 1 } ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
69		2wlklem 23614	. . . . . . . . . . . . . . . . 17 |- ( A. i e. { 0 , 1 } ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
70	68, 69	syl6bb 261	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> ( A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) )
71	66, 70	anbi12d 710	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 2 -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) <-> ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
72	2	imbi1d 317	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 2 -> ( ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) <-> ( P : ( 0 ... 2 ) --> V -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) ) )
73	71, 72	imbi12d 320	. . . . . . . . . . . . . 14 |- ( ( # ` F ) = 2 -> ( ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) ) <-> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P : ( 0 ... 2 ) --> V -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) ) ) )
74	73	ad2antll 728	. . . . . . . . . . . . 13 |- ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) -> ( ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) ) <-> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( P : ( 0 ... 2 ) --> V -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) ) ) )
75	63, 74	mpbird 232	. . . . . . . . . . . 12 |- ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) ) )
76	75	imp 429	. . . . . . . . . . 11 |- ( ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) )
77	76	impcom 430	. . . . . . . . . 10 |- ( ( P : ( 0 ... ( # ` F ) ) --> V /\ ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) )
78	9, 77	jca 532	. . . . . . . . 9 |- ( ( P : ( 0 ... ( # ` F ) ) --> V /\ ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) -> ( P : { 0 , 1 , 2 } --> V /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) )
79		2z 10788	. . . . . . . . . . . 12 |- 2 e. ZZ
80	22, 40, 79	3pm3.2i 1166	. . . . . . . . . . 11 |- ( 0 e. _V /\ 1 e. _V /\ 2 e. ZZ )
81		0ne1 10499	. . . . . . . . . . . 12 |- 0 =/= 1
82		0ne2 10643	. . . . . . . . . . . 12 |- 0 =/= 2
83		1ne2 10644	. . . . . . . . . . . 12 |- 1 =/= 2
84	81, 82, 83	3pm3.2i 1166	. . . . . . . . . . 11 |- ( 0 =/= 1 /\ 0 =/= 2 /\ 1 =/= 2 )
85	80, 84	pm3.2i 455	. . . . . . . . . 10 |- ( ( 0 e. _V /\ 1 e. _V /\ 2 e. ZZ ) /\ ( 0 =/= 1 /\ 0 =/= 2 /\ 1 =/= 2 ) )
86		eqid 2454	. . . . . . . . . . 11 |- { 0 , 1 , 2 } = { 0 , 1 , 2 }
87	86	f13dfv 30294	. . . . . . . . . 10 |- ( ( ( 0 e. _V /\ 1 e. _V /\ 2 e. ZZ ) /\ ( 0 =/= 1 /\ 0 =/= 2 /\ 1 =/= 2 ) ) -> ( P : { 0 , 1 , 2 } -1-1-> V <-> ( P : { 0 , 1 , 2 } --> V /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) ) )
88	85, 87	mp1i 12	. . . . . . . . 9 |- ( ( P : ( 0 ... ( # ` F ) ) --> V /\ ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) -> ( P : { 0 , 1 , 2 } -1-1-> V <-> ( P : { 0 , 1 , 2 } --> V /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) ) )
89	78, 88	mpbird 232	. . . . . . . 8 |- ( ( P : ( 0 ... ( # ` F ) ) --> V /\ ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) -> P : { 0 , 1 , 2 } -1-1-> V )
90		df-f1 5530	. . . . . . . 8 |- ( P : { 0 , 1 , 2 } -1-1-> V <-> ( P : { 0 , 1 , 2 } --> V /\ Fun `' P ) )
91	89, 90	sylib 196	. . . . . . 7 |- ( ( P : ( 0 ... ( # ` F ) ) --> V /\ ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) -> ( P : { 0 , 1 , 2 } --> V /\ Fun `' P ) )
92	91	simprd 463	. . . . . 6 |- ( ( P : ( 0 ... ( # ` F ) ) --> V /\ ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) -> Fun `' P )
93	92	expcom 435	. . . . 5 |- ( ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> Fun `' P ) )
94	93	ex 434	. . . 4 |- ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> Fun `' P ) ) )
95	94	com23 78	. . 3 |- ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' P ) ) )
96	95	impancom 440	. 2 |- ( ( V USGrph E /\ P : ( 0 ... ( # ` F ) ) --> V ) -> ( ( F e. Word dom E /\ ( # ` F ) = 2 ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' P ) ) )
97	96	impcom 430	1 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ ( V USGrph E /\ P : ( 0 ... ( # ` F ) ) --> V ) ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' P ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2647   A.wral 2798   _Vcvv 3076   {cpr 3986   {ctp 3988   class class class wbr 4399   `'ccnv 4946   dom cdm 4947   ran crn 4948   Fun wfun 5519   -->wf 5521   -1-1->wf1 5522   ` cfv 5525  (class class class)co 6199   0cc0 9392   1c1 9393   + caddc 9395   2c2 10481   ZZcz 10756   ...cfz 11553  ..^cfzo 11664   #chash 12219  Word cword 12338   USGrph cusg 23415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-hash 12220  df-word 12346  df-usgra 23417

This theorem is referenced by:  usgra2wlkspth  30445