Theorem usgra2wlkspthlem2 24598

Description: Lemma 2 for usgra2wlkspth 24599. (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 6289	. . . . . . . . . . . . . 14 |- ( ( # ` F ) = 2 -> ( 0 ... ( # ` F ) ) = ( 0 ... 2 ) )
2	1	feq2d 5708	. . . . . . . . . . . . 13 |- ( ( # ` F ) = 2 -> ( P : ( 0 ... ( # ` F ) ) --> V <-> P : ( 0 ... 2 ) --> V ) )
3		fz0tp 11788	. . . . . . . . . . . . . . 15 |- ( 0 ... 2 ) = { 0 , 1 , 2 }
4	3	feq2i 5714	. . . . . . . . . . . . . 14 |- ( P : ( 0 ... 2 ) --> V <-> P : { 0 , 1 , 2 } --> V )
5	4	biimpi 194	. . . . . . . . . . . . 13 |- ( P : ( 0 ... 2 ) --> V -> P : { 0 , 1 , 2 } --> V )
6	2, 5	syl6bi 228	. . . . . . . . . . . 12 |- ( ( # ` F ) = 2 -> ( P : ( 0 ... ( # ` F ) ) --> V -> P : { 0 , 1 , 2 } --> V ) )
7	6	ad2antll 728	. . . . . . . . . . 11 |- ( ( V USGrph E /\ ( F e. Word dom E /\ ( # ` F ) = 2 ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> P : { 0 , 1 , 2 } --> V ) )
8	7	adantr 465	. . . . . . . . . 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 430	. . . . . . . . 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 2462	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( P ` 0 ) = ( P ` 2 ) <-> A = B ) )
11	10	biimpd 207	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( P ` 0 ) = ( P ` 2 ) -> A = B ) )
12	11	necon3d 2667	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( A =/= B -> ( P ` 0 ) =/= ( P ` 2 ) ) )
13	12	3impia 1194	. . . . . . . . . . . . . . . . . 18 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) -> ( P ` 0 ) =/= ( P ` 2 ) )
14	13	adantr 465	. . . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . 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 755	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) -> V USGrph E )
18		usgrafun 24327	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( V USGrph E -> Fun E )
19		wrdf 12535	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( F e. Word dom E -> F : ( 0..^ ( # ` F ) ) --> dom E )
20		oveq2 6289	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = ( 0..^ 2 ) )
21	20	feq2d 5708	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E <-> F : ( 0..^ 2 ) --> dom E ) )
22		c0ex 9593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 0 e. _V
23	22	prid1 4123	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- 0 e. { 0 , 1 }
24		fzo0to2pr 11881	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( 0..^ 2 ) = { 0 , 1 }
25	23, 24	eleqtrri 2530	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- 0 e. ( 0..^ 2 )
26		ffvelrn 6014	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( F : ( 0..^ 2 ) --> dom E /\ 0 e. ( 0..^ 2 ) ) -> ( F ` 0 ) e. dom E )
27	25, 26	mpan2 671	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( F : ( 0..^ 2 ) --> dom E -> ( F ` 0 ) e. dom E )
28	21, 27	syl6bi 228	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( F ` 0 ) e. dom E ) )
29	19, 28	mpan9 469	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( F e. Word dom E /\ ( # ` F ) = 2 ) -> ( F ` 0 ) e. dom E )
30		fvelrn 6009	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( Fun E /\ ( F ` 0 ) e. dom E ) -> ( E ` ( F ` 0 ) ) e. ran E )
31	18, 29, 30	syl2anr 478	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( E ` ( F ` 0 ) ) e. ran E )
32	31	adantr 465	. . . . . . . . . . . . . . . . . . . . . 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 2515	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } -> ( ( E ` ( F ` 0 ) ) e. ran E <-> { ( P ` 0 ) , ( P ` 1 ) } e. ran E ) )
34	33	adantl 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 <-> { ( P ` 0 ) , ( P ` 1 ) } e. ran E ) )
35	32, 34	mpbid 210	. . . . . . . . . . . . . . . . . . . . 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 24359	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( V USGrph E /\ { ( P ` 0 ) , ( P ` 1 ) } e. ran E ) -> ( P ` 0 ) =/= ( P ` 1 ) )
37	17, 35, 36	syl2anc 661	. . . . . . . . . . . . . . . . . . . 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 434	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } -> ( P ` 0 ) =/= ( P ` 1 ) ) )
39		simplr 755	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> V USGrph E )
40		1ex 9594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 1 e. _V
41	40	prid2 4124	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- 1 e. { 0 , 1 }
42	41, 24	eleqtrri 2530	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- 1 e. ( 0..^ 2 )
43		ffvelrn 6014	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( F : ( 0..^ 2 ) --> dom E /\ 1 e. ( 0..^ 2 ) ) -> ( F ` 1 ) e. dom E )
44	42, 43	mpan2 671	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( F : ( 0..^ 2 ) --> dom E -> ( F ` 1 ) e. dom E )
45	21, 44	syl6bi 228	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( F ` 1 ) e. dom E ) )
46	19, 45	mpan9 469	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( F e. Word dom E /\ ( # ` F ) = 2 ) -> ( F ` 1 ) e. dom E )
47		fvelrn 6009	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( Fun E /\ ( F ` 1 ) e. dom E ) -> ( E ` ( F ` 1 ) ) e. ran E )
48	18, 46, 47	syl2anr 478	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( F e. Word dom E /\ ( # ` F ) = 2 ) /\ V USGrph E ) -> ( E ` ( F ` 1 ) ) e. ran E )
49	48	adantr 465	. . . . . . . . . . . . . . . . . . . . . 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 2515	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } -> ( ( E ` ( F ` 1 ) ) e. ran E <-> { ( P ` 1 ) , ( P ` 2 ) } e. ran E ) )
51	50	adantl 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 <-> { ( P ` 1 ) , ( P ` 2 ) } e. ran E ) )
52	49, 51	mpbid 210	. . . . . . . . . . . . . . . . . . . . 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 24359	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( V USGrph E /\ { ( P ` 1 ) , ( P ` 2 ) } e. ran E ) -> ( P ` 1 ) =/= ( P ` 2 ) )
54	39, 52, 53	syl2anc 661	. . . . . . . . . . . . . . . . . . . 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 434	. . . . . . . . . . . . . . . . . . 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 563	. . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . 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 429	. . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . 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 987	. . . . . . . . . . . . . . 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 664	. . . . . . . . . . . . . 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 610	. . . . . . . . . . . . 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 430	. . . . . . . . . . . 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 5856	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = ( P ` 2 ) )
65	64	eqeq1d 2445	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> ( ( P ` ( # ` F ) ) = B <-> ( P ` 2 ) = B ) )
66	65	3anbi2d 1305	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 2 -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) <-> ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) ) )
67	20, 24	syl6eq 2500	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = { 0 , 1 } )
68	67	raleqdv 3046	. . . . . . . . . . . . . . . 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 24544	. . . . . . . . . . . . . . . 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 261	. . . . . . . . . . . . . . 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 710	. . . . . . . . . . . . . 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 317	. . . . . . . . . . . . . 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 320	. . . . . . . . . . . . 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 728	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 429	. . . . . . . . . 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 430	. . . . . . . . 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 10903	. . . . . . . . . . . 12 |- 2 e. ZZ
79	22, 40, 78	3pm3.2i 1175	. . . . . . . . . . 11 |- ( 0 e. _V /\ 1 e. _V /\ 2 e. ZZ )
80		0ne1 10610	. . . . . . . . . . . 12 |- 0 =/= 1
81		0ne2 10754	. . . . . . . . . . . 12 |- 0 =/= 2
82		1ne2 10755	. . . . . . . . . . . 12 |- 1 =/= 2
83	80, 81, 82	3pm3.2i 1175	. . . . . . . . . . 11 |- ( 0 =/= 1 /\ 0 =/= 2 /\ 1 =/= 2 )
84	79, 83	pm3.2i 455	. . . . . . . . . 10 |- ( ( 0 e. _V /\ 1 e. _V /\ 2 e. ZZ ) /\ ( 0 =/= 1 /\ 0 =/= 2 /\ 1 =/= 2 ) )
85		eqid 2443	. . . . . . . . . . 11 |- { 0 , 1 , 2 } = { 0 , 1 , 2 }
86	85	f13dfv 6165	. . . . . . . . . 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 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 ) ) ) ) )
88	9, 77, 87	mpbir2and 922	. . . . . . . 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 5583	. . . . . . . 8 |- ( P : { 0 , 1 , 2 } -1-1-> V <-> ( P : { 0 , 1 , 2 } --> V /\ Fun `' P ) )
90	88, 89	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 ) )
91	90	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 )
92	91	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 ) )
93	92	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 ) ) )
94	93	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 ) ) )
95	94	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 ) ) )
96	95	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 974   = wceq 1383   e. wcel 1804   =/= wne 2638   A.wral 2793   _Vcvv 3095   {cpr 4016   {ctp 4018   class class class wbr 4437   `'ccnv 4988   dom cdm 4989   ran crn 4990   Fun wfun 5572   -->wf 5574   -1-1->wf1 5575   ` cfv 5578  (class class class)co 6281   0cc0 9495   1c1 9496   + caddc 9498   2c2 10592   ZZcz 10871   ...cfz 11683  ..^cfzo 11806   #chash 12387  Word cword 12516   USGrph cusg 24308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-hash 12388  df-word 12524  df-usgra 24311

This theorem is referenced by:  usgra2wlkspth  24599