Theorem usgra2wlkspthlem1 25426

Description: Lemma 1 for usgra2wlkspth 25428. (Contributed by Alexander van der Vekens, 2-Mar-2018.)

Assertion

Ref	Expression
usgra2wlkspthlem1	|- ( ( F e. Word dom E /\ E : dom E -1-1-> ran 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 `' F ) )

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

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

Proof of Theorem usgra2wlkspthlem1

Step	Hyp	Ref	Expression
1		wrdf 12723	. . 3 |- ( F e. Word dom E -> F : ( 0..^ ( # ` F ) ) --> dom E )
2		simpl 464	. . . . . . . . . . . . 13 |- ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> F : ( 0..^ ( # ` F ) ) --> dom E )
3	2	adantr 472	. . . . . . . . . . . 12 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> F : ( 0..^ ( # ` F ) ) --> dom E )
4		oveq2 6316	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = ( 0..^ 2 ) )
5		fzo0to2pr 12027	. . . . . . . . . . . . . . . 16 |- ( 0..^ 2 ) = { 0 , 1 }
6	4, 5	syl6req 2522	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 2 -> { 0 , 1 } = ( 0..^ ( # ` F ) ) )
7	6	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> { 0 , 1 } = ( 0..^ ( # ` F ) ) )
8	7	ad2antlr 741	. . . . . . . . . . . . 13 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> { 0 , 1 } = ( 0..^ ( # ` F ) ) )
9	8	feq2d 5725	. . . . . . . . . . . 12 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( F : { 0 , 1 } --> dom E <-> F : ( 0..^ ( # ` F ) ) --> dom E ) )
10	3, 9	mpbird 240	. . . . . . . . . . 11 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> F : { 0 , 1 } --> dom E )
11		preq1 4042	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 0 ) = A -> { ( P ` 0 ) , ( P ` 1 ) } = { A , ( P ` 1 ) } )
12	11	eqeq2d 2481	. . . . . . . . . . . . . . . . . . 19 |- ( ( P ` 0 ) = A -> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } <-> ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } ) )
13		preq2 4043	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 2 ) = B -> { ( P ` 1 ) , ( P ` 2 ) } = { ( P ` 1 ) , B } )
14	13	eqeq2d 2481	. . . . . . . . . . . . . . . . . . 19 |- ( ( P ` 2 ) = B -> ( ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } <-> ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) )
15	12, 14	bi2anan9 890	. . . . . . . . . . . . . . . . . 18 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) <-> ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) )
16	15	3adant3 1050	. . . . . . . . . . . . . . . . 17 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) -> ( ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) <-> ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) )
17		fveq2 5879	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( E ` ( F ` 0 ) ) = ( E ` ( F ` 1 ) ) )
18	17	eqeq1d 2473	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } <-> ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } ) )
19	18	anbi1d 719	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) <-> ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) )
20	19	anbi2d 718	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) <-> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) ) )
21		eqtr2 2491	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> { A , ( P ` 1 ) } = { ( P ` 1 ) , B } )
22		fvex 5889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( P ` 0 ) e. _V
23		eleq1 2537	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( A = ( P ` 0 ) -> ( A e. _V <-> ( P ` 0 ) e. _V ) )
24	23	eqcoms 2479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( P ` 0 ) = A -> ( A e. _V <-> ( P ` 0 ) e. _V ) )
25	22, 24	mpbiri 241	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( P ` 0 ) = A -> A e. _V )
26		fvex 5889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( P ` 1 ) e. _V
27	26	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( P ` 0 ) = A -> ( P ` 1 ) e. _V )
28	25, 27	jca 541	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( P ` 0 ) = A -> ( A e. _V /\ ( P ` 1 ) e. _V ) )
29	26	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( P ` 2 ) = B -> ( P ` 1 ) e. _V )
30		fvex 5889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( P ` 2 ) e. _V
31		eleq1 2537	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( B = ( P ` 2 ) -> ( B e. _V <-> ( P ` 2 ) e. _V ) )
32	31	eqcoms 2479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( P ` 2 ) = B -> ( B e. _V <-> ( P ` 2 ) e. _V ) )
33	30, 32	mpbiri 241	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( P ` 2 ) = B -> B e. _V )
34	29, 33	jca 541	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( P ` 2 ) = B -> ( ( P ` 1 ) e. _V /\ B e. _V ) )
35	28, 34	anim12i 576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( A e. _V /\ ( P ` 1 ) e. _V ) /\ ( ( P ` 1 ) e. _V /\ B e. _V ) ) )
36	35	adantr 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> ( ( A e. _V /\ ( P ` 1 ) e. _V ) /\ ( ( P ` 1 ) e. _V /\ B e. _V ) ) )
37		preq12bg 4146	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( A e. _V /\ ( P ` 1 ) e. _V ) /\ ( ( P ` 1 ) e. _V /\ B e. _V ) ) -> ( { A , ( P ` 1 ) } = { ( P ` 1 ) , B } <-> ( ( A = ( P ` 1 ) /\ ( P ` 1 ) = B ) \/ ( A = B /\ ( P ` 1 ) = ( P ` 1 ) ) ) ) )
38	36, 37	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> ( { A , ( P ` 1 ) } = { ( P ` 1 ) , B } <-> ( ( A = ( P ` 1 ) /\ ( P ` 1 ) = B ) \/ ( A = B /\ ( P ` 1 ) = ( P ` 1 ) ) ) ) )
39		eqtr 2490	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( A = ( P ` 1 ) /\ ( P ` 1 ) = B ) -> A = B )
40		simpl 464	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( A = B /\ ( P ` 1 ) = ( P ` 1 ) ) -> A = B )
41	39, 40	jaoi 386	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( A = ( P ` 1 ) /\ ( P ` 1 ) = B ) \/ ( A = B /\ ( P ` 1 ) = ( P ` 1 ) ) ) -> A = B )
42	38, 41	syl6bi 236	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> ( { A , ( P ` 1 ) } = { ( P ` 1 ) , B } -> A = B ) )
43	21, 42	syl5com 30	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> A = B ) )
44	43	impcom 437	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) -> A = B )
45	20, 44	syl6bi 236	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) -> A = B ) )
46	45	com12 31	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) -> ( ( F ` 0 ) = ( F ` 1 ) -> A = B ) )
47	46	necon3d 2664	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) -> ( A =/= B -> ( F ` 0 ) =/= ( F ` 1 ) ) )
48	47	a1d 25	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) -> ( F : ( 0..^ 2 ) --> dom E -> ( A =/= B -> ( F ` 0 ) =/= ( F ` 1 ) ) ) )
49	48	exp31 615	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> ( F : ( 0..^ 2 ) --> dom E -> ( A =/= B -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) )
50	49	com25 93	. . . . . . . . . . . . . . . . . 18 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( A =/= B -> ( ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) )
51	50	3impia 1228	. . . . . . . . . . . . . . . . 17 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) -> ( ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
52	16, 51	sylbid 223	. . . . . . . . . . . . . . . 16 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) -> ( ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
53	52	imp 436	. . . . . . . . . . . . . . 15 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) )
54	53	com13 82	. . . . . . . . . . . . . 14 |- ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) )
55	4	feq2d 5725	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E <-> F : ( 0..^ 2 ) --> dom E ) )
56		fveq2 5879	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = ( P ` 2 ) )
57	56	eqeq1d 2473	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` F ) = 2 -> ( ( P ` ( # ` F ) ) = B <-> ( P ` 2 ) = B ) )
58	57	3anbi2d 1370	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` F ) = 2 -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) <-> ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) ) )
59	4, 5	syl6eq 2521	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = { 0 , 1 } )
60	59	raleqdv 2979	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` 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 ) ) } ) )
61		2wlklem 25373	. . . . . . . . . . . . . . . . . . 19 |- ( 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 ) } ) )
62	60, 61	syl6bb 269	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` 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 ) } ) ) )
63	58, 62	anbi12d 725	. . . . . . . . . . . . . . . . 17 |- ( ( # ` 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 ) } ) ) ) )
64	63	imbi1d 324	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> ( ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) <-> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) )
65	55, 64	imbi12d 327	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 2 -> ( ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) <-> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
66	65	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) <-> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
67	54, 66	mpbird 240	. . . . . . . . . . . . 13 |- ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) )
68	67	impcom 437	. . . . . . . . . . . 12 |- ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) )
69	68	imp 436	. . . . . . . . . . 11 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( F ` 0 ) =/= ( F ` 1 ) )
70	10, 69	jca 541	. . . . . . . . . 10 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( F : { 0 , 1 } --> dom E /\ ( F ` 0 ) =/= ( F ` 1 ) ) )
71		c0ex 9655	. . . . . . . . . . . . 13 |- 0 e. _V
72		1ex 9656	. . . . . . . . . . . . 13 |- 1 e. _V
73	71, 72	pm3.2i 462	. . . . . . . . . . . 12 |- ( 0 e. _V /\ 1 e. _V )
74		0ne1 10699	. . . . . . . . . . . 12 |- 0 =/= 1
75	73, 74	pm3.2i 462	. . . . . . . . . . 11 |- ( ( 0 e. _V /\ 1 e. _V ) /\ 0 =/= 1 )
76		eqid 2471	. . . . . . . . . . . 12 |- { 0 , 1 } = { 0 , 1 }
77	76	f12dfv 6190	. . . . . . . . . . 11 |- ( ( ( 0 e. _V /\ 1 e. _V ) /\ 0 =/= 1 ) -> ( F : { 0 , 1 } -1-1-> dom E <-> ( F : { 0 , 1 } --> dom E /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) )
78	75, 77	mp1i 13	. . . . . . . . . 10 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( F : { 0 , 1 } -1-1-> dom E <-> ( F : { 0 , 1 } --> dom E /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) )
79	70, 78	mpbird 240	. . . . . . . . 9 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> F : { 0 , 1 } -1-1-> dom E )
80		df-f1 5594	. . . . . . . . 9 |- ( F : { 0 , 1 } -1-1-> dom E <-> ( F : { 0 , 1 } --> dom E /\ Fun `' F ) )
81	79, 80	sylib 201	. . . . . . . 8 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( F : { 0 , 1 } --> dom E /\ Fun `' F ) )
82	81	simprd 470	. . . . . . 7 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> Fun `' F )
83	82	ex 441	. . . . . 6 |- ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' F ) )
84	83	expcom 442	. . . . 5 |- ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' F ) ) )
85	84	ex 441	. . . 4 |- ( ( # ` F ) = 2 -> ( E : dom E -1-1-> ran E -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' F ) ) ) )
86	85	com13 82	. . 3 |- ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( E : dom E -1-1-> ran 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 `' F ) ) ) )
87	1, 86	syl 17	. 2 |- ( F e. Word dom E -> ( E : dom E -1-1-> ran 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 `' F ) ) ) )
88	87	3imp 1224	1 |- ( ( F e. Word dom E /\ E : dom E -1-1-> ran 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 `' F ) )

Syntax hints:   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   _Vcvv 3031   {cpr 3961   `'ccnv 4838   dom cdm 4839   ran crn 4840   Fun wfun 5583   -->wf 5585   -1-1->wf1 5586   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558   + caddc 9560   2c2 10681  ..^cfzo 11942   #chash 12553  Word cword 12703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711

This theorem is referenced by:  usgra2wlkspth  25428