Theorem usgra2wlkspthlem2 25346

Description: Lemma 2 for usgra2wlkspth 25347. (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 6313	. . . . . . . . . . . . . 14 |- ( ( # ` F ) = 2 -> ( 0 ... ( # ` F ) ) = ( 0 ... 2 ) )
2	1	feq2d 5733	. . . . . . . . . . . . 13 |- ( ( # ` F ) = 2 -> ( P : ( 0 ... ( # ` F ) ) --> V <-> P : ( 0 ... 2 ) --> V ) )
3		fz0tp 11900	. . . . . . . . . . . . . . 15 |- ( 0 ... 2 ) = { 0 , 1 , 2 }
4	3	feq2i 5739	. . . . . . . . . . . . . 14 |- ( P : ( 0 ... 2 ) --> V <-> P : { 0 , 1 , 2 } --> V )
5	4	biimpi 197	. . . . . . . . . . . . 13 |- ( P : ( 0 ... 2 ) --> V -> P : { 0 , 1 , 2 } --> V )
6	2, 5	syl6bi 231	. . . . . . . . . . . 12 |- ( ( # ` F ) = 2 -> ( P : ( 0 ... ( # ` F ) ) --> V -> P : { 0 , 1 , 2 } --> V ) )
7	6	ad2antll 733	. . . . . . . . . . 11 |- ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> P : { 0 , 1 , 2 } --> V ) )
8	7	adantr 466	. . . . . . . . . 10 |- ( ( ( 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 431	. . . . . . . . 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 )
10		eqeq12 2441	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( P ` 0 ) = ( P ` 2 ) <-> A = B ) )
11	10	biimpd 210	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( P ` 0 ) = ( P ` 2 ) -> A = B ) )
12	11	necon3d 2644	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( A =/= B -> ( P ` 0 ) =/= ( P ` 2 ) ) )
13	12	3impia 1202	. . . . . . . . . . . . . . . . . 18 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) -> ( P ` 0 ) =/= ( P ` 2 ) )
14	13	adantr 466	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( 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 467	. . . . . . . . . . . . . . . 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 ) =/= ( P ` 2 ) )
16	15	adantr 466	. . . . . . . . . . . . . . 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 ` 2 ) )
17		simplr 760	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) -> V USGrph E )
18		usgrafun 25074	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( V USGrph E -> Fun E )
19		wrdf 12680	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( F e. Word dom E -> F : ( 0..^ ( # ` F ) ) --> dom E )
20		oveq2 6313	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = ( 0..^ 2 ) )
21	20	feq2d 5733	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E <-> F : ( 0..^ 2 ) --> dom E ) )
22		c0ex 9644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 0 e. _V
23	22	prid1 4108	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- 0 e. { 0 , 1 }
24		fzo0to2pr 12004	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( 0..^ 2 ) = { 0 , 1 }
25	23, 24	eleqtrri 2506	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- 0 e. ( 0..^ 2 )
26		ffvelrn 6035	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( F : ( 0..^ 2 ) --> dom E /\ 0 e. ( 0..^ 2 ) ) -> ( F ` 0 ) e. dom E )
27	25, 26	mpan2 675	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( F : ( 0..^ 2 ) --> dom E -> ( F ` 0 ) e. dom E )
28	21, 27	syl6bi 231	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( F ` 0 ) e. dom E ) )
29	19, 28	mpan9 471	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( F e. Word dom E /\ ( # ` F ) = 2 ) -> ( F ` 0 ) e. dom E )
30		fvelrn 6030	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( Fun E /\ ( F ` 0 ) e. dom E ) -> ( E ` ( F ` 0 ) ) e. ran E )
31	18, 29, 30	syl2anr 480	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( E ` ( F ` 0 ) ) e. ran E )
32	31	adantr 466	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( 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 2495	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } -> ( ( E ` ( F ` 0 ) ) e. ran E <-> { ( P ` 0 ) , ( P ` 1 ) } e. ran E ) )
34	33	adantl 467	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( 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 213	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( 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 25106	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( V USGrph E /\ { ( P ` 0 ) , ( P ` 1 ) } e. ran E ) -> ( P ` 0 ) =/= ( P ` 1 ) )
37	17, 35, 36	syl2anc 665	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) -> ( P ` 0 ) =/= ( P ` 1 ) )
38	37	ex 435	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } -> ( P ` 0 ) =/= ( P ` 1 ) ) )
39		simplr 760	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> V USGrph E )
40		1ex 9645	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 1 e. _V
41	40	prid2 4109	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- 1 e. { 0 , 1 }
42	41, 24	eleqtrri 2506	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- 1 e. ( 0..^ 2 )
43		ffvelrn 6035	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( F : ( 0..^ 2 ) --> dom E /\ 1 e. ( 0..^ 2 ) ) -> ( F ` 1 ) e. dom E )
44	42, 43	mpan2 675	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( F : ( 0..^ 2 ) --> dom E -> ( F ` 1 ) e. dom E )
45	21, 44	syl6bi 231	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( F ` 1 ) e. dom E ) )
46	19, 45	mpan9 471	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( F e. Word dom E /\ ( # ` F ) = 2 ) -> ( F ` 1 ) e. dom E )
47		fvelrn 6030	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( Fun E /\ ( F ` 1 ) e. dom E ) -> ( E ` ( F ` 1 ) ) e. ran E )
48	18, 46, 47	syl2anr 480	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( E ` ( F ` 1 ) ) e. ran E )
49	48	adantr 466	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( 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 2495	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } -> ( ( E ` ( F ` 1 ) ) e. ran E <-> { ( P ` 1 ) , ( P ` 2 ) } e. ran E ) )
51	50	adantl 467	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( 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 213	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( 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 25106	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( V USGrph E /\ { ( P ` 1 ) , ( P ` 2 ) } e. ran E ) -> ( P ` 1 ) =/= ( P ` 2 ) )
54	39, 52, 53	syl2anc 665	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( P ` 1 ) =/= ( P ` 2 ) )
55	54	ex 435	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( 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 565	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 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 468	. . . . . . . . . . . . . . . . 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 ) ) ) )
58	57	imp 430	. . . . . . . . . . . . . . . 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 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) )
59	58	adantr 466	. . . . . . . . . . . . . . 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 ` 1 ) =/= ( P ` 2 ) ) )
60		3anan12 995	. . . . . . . . . . . . . . 15 |- ( ( ( 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 668	. . . . . . . . . . . . . 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 ) ) )
62	61	exp41 613	. . . . . . . . . . . . 13 |- ( ( 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 431	. . . . . . . . . . . 12 |- ( ( 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 5881	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = ( P ` 2 ) )
65	64	eqeq1d 2424	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> ( ( P ` ( # ` F ) ) = B <-> ( P ` 2 ) = B ) )
66	65	3anbi2d 1340	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 2 -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) <-> ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) ) )
67	20, 24	syl6eq 2479	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = { 0 , 1 } )
68	67	raleqdv 3028	. . . . . . . . . . . . . . . 16 |- ( ( # ` 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 25292	. . . . . . . . . . . . . . . 16 |- ( 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 264	. . . . . . . . . . . . . . 15 |- ( ( # ` 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 715	. . . . . . . . . . . . . 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 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
72	2	imbi1d 318	. . . . . . . . . . . . . 14 |- ( ( # ` 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 321	. . . . . . . . . . . . 13 |- ( ( # ` 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 733	. . . . . . . . . . . 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 ) ) ) ) <-> ( ( ( ( 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 235	. . . . . . . . . . 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 ) ) ) ) )
76	75	imp 430	. . . . . . . . . 10 |- ( ( ( 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 431	. . . . . . . . 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 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) )
78		2z 10976	. . . . . . . . . . . 12 |- 2 e. ZZ
79	22, 40, 78	3pm3.2i 1183	. . . . . . . . . . 11 |- ( 0 e. _V /\ 1 e. _V /\ 2 e. ZZ )
80		0ne1 10684	. . . . . . . . . . . 12 |- 0 =/= 1
81		0ne2 10828	. . . . . . . . . . . 12 |- 0 =/= 2
82		1ne2 10829	. . . . . . . . . . . 12 |- 1 =/= 2
83	80, 81, 82	3pm3.2i 1183	. . . . . . . . . . 11 |- ( 0 =/= 1 /\ 0 =/= 2 /\ 1 =/= 2 )
84	79, 83	pm3.2i 456	. . . . . . . . . 10 |- ( ( 0 e. _V /\ 1 e. _V /\ 2 e. ZZ ) /\ ( 0 =/= 1 /\ 0 =/= 2 /\ 1 =/= 2 ) )
85		eqid 2422	. . . . . . . . . . 11 |- { 0 , 1 , 2 } = { 0 , 1 , 2 }
86	85	f13dfv 6188	. . . . . . . . . 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 ) ) ) ) )
87	84, 86	mp1i 13	. . . . . . . . 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 ) ) ) ) )
88	9, 77, 87	mpbir2and 930	. . . . . . . 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 )
89		df-f1 5606	. . . . . . . 8 |- ( P : { 0 , 1 , 2 } -1-1-> V <-> ( P : { 0 , 1 , 2 } --> V /\ Fun `' P ) )
90	88, 89	sylib 199	. . . . . . 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 ) )
91	90	simprd 464	. . . . . 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 )
92	91	expcom 436	. . . . 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 ) )
93	92	ex 435	. . . 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 ) ) )
94	93	com23 81	. . 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 ) ) )
95	94	impancom 441	. 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 ) ) )
96	95	impcom 431	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 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1872   =/= wne 2614   A.wral 2771   _Vcvv 3080   {cpr 4000   {ctp 4002   class class class wbr 4423   `'ccnv 4852   dom cdm 4853   ran crn 4854   Fun wfun 5595   -->wf 5597   -1-1->wf1 5598   ` cfv 5601  (class class class)co 6305   0cc0 9546   1c1 9547   + caddc 9549   2c2 10666   ZZcz 10944   ...cfz 11791  ..^cfzo 11922   #chash 12521  Word cword 12660   USGrph cusg 25055

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-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-usgra 25058

This theorem is referenced by:  usgra2wlkspth  25347