Theorem upgr2wlkdc 16147

Description: Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 28-Oct-2021.)

Hypotheses

Ref	Expression
upgr2wlk.v	|- V = (Vtx ` G )
upgr2wlk.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
upgr2wlkdc	|- ( G e. UPGraph -> ( ( F (Walks ` G ) P /\ F ~~ 2o ) <-> ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )

Distinct variable groups:   k, F   P, k   k, G   k, I   k, V

Proof of Theorem upgr2wlkdc

Step	Hyp	Ref	Expression
1		simprl 529	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> F (Walks ` G ) P )
2		upgr2wlk.v	. . . . . . . . . 10 |- V = (Vtx ` G )
3		upgr2wlk.i	. . . . . . . . . 10 |- I = (iEdg ` G )
4	2, 3	upgriswlkdc 16132	. . . . . . . . 9 |- ( G e. UPGraph -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) )
5	4	adantr 276	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) )
6	1, 5	mpbid 147	. . . . . . 7 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
7	6	simp1d 1033	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> F e. Word dom I )
8		wrdf 11095	. . . . . 6 |- ( F e. Word dom I -> F : ( 0..^ (♯ ` F ) ) --> dom I )
9	7, 8	syl 14	. . . . 5 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> F : ( 0..^ (♯ ` F ) ) --> dom I )
10		simprr 531	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> F ~~ 2o )
11		hash2en 11083	. . . . . . . . 9 |- ( F ~~ 2o <-> ( F e. Fin /\ (♯ ` F ) = 2 ) )
12	11	simprbi 275	. . . . . . . 8 |- ( F ~~ 2o -> (♯ ` F ) = 2 )
13	10, 12	syl 14	. . . . . . 7 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> (♯ ` F ) = 2 )
14	13	oveq2d 6026	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( 0..^ (♯ ` F ) ) = ( 0..^ 2 ) )
15	14	feq2d 5464	. . . . 5 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( F : ( 0..^ (♯ ` F ) ) --> dom I <-> F : ( 0..^ 2 ) --> dom I ) )
16	9, 15	mpbid 147	. . . 4 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> F : ( 0..^ 2 ) --> dom I )
17	6	simp2d 1034	. . . . 5 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> P : ( 0 ... (♯ ` F ) ) --> V )
18	13	oveq2d 6026	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( 0 ... (♯ ` F ) ) = ( 0 ... 2 ) )
19	18	feq2d 5464	. . . . 5 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( P : ( 0 ... (♯ ` F ) ) --> V <-> P : ( 0 ... 2 ) --> V ) )
20	17, 19	mpbid 147	. . . 4 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> P : ( 0 ... 2 ) --> V )
21	6	simp3d 1035	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
22		simpl 109	. . . . . . 7 |- ( (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> DECID ( P ` k ) = ( P ` ( k + 1 ) ) )
23	22	ralimi 2593	. . . . . 6 |- ( A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> A. k e. ( 0..^ (♯ ` F ) )DECID ( P ` k ) = ( P ` ( k + 1 ) ) )
24	21, 23	syl 14	. . . . 5 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> A. k e. ( 0..^ (♯ ` F ) )DECID ( P ` k ) = ( P ` ( k + 1 ) ) )
25		oveq2 6018	. . . . . . 7 |- ( (♯ ` F ) = 2 -> ( 0..^ (♯ ` F ) ) = ( 0..^ 2 ) )
26		fzo0to2pr 10441	. . . . . . 7 |- ( 0..^ 2 ) = { 0 , 1 }
27	25, 26	eqtrdi 2278	. . . . . 6 |- ( (♯ ` F ) = 2 -> ( 0..^ (♯ ` F ) ) = { 0 , 1 } )
28	13, 27	syl 14	. . . . 5 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( 0..^ (♯ ` F ) ) = { 0 , 1 } )
29	24, 28	raleqtrdv 2736	. . . 4 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) )
30	16, 20, 29	3jca 1201	. . 3 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) )
31		simpr 110	. . . . . 6 |- ( (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
32	31	ralimi 2593	. . . . 5 |- ( A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
33	21, 32	syl 14	. . . 4 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
34	28	raleqdv 2734	. . . . 5 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> A. k e. { 0 , 1 } ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
35		2wlklem 16146	. . . . 5 |- ( A. k e. { 0 , 1 } ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
36	34, 35	bitrdi 196	. . . 4 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) )
37	33, 36	mpbid 147	. . 3 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
38	30, 37	jca 306	. 2 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) )
39		simprl1 1066	. . . . . 6 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> F : ( 0..^ 2 ) --> dom I )
40		2nn0 9402	. . . . . 6 |- 2 e. NN0
41		iswrdinn0 11094	. . . . . 6 |- ( ( F : ( 0..^ 2 ) --> dom I /\ 2 e. NN0 ) -> F e. Word dom I )
42	39, 40, 41	sylancl 413	. . . . 5 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> F e. Word dom I )
43		simprl2 1067	. . . . . 6 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> P : ( 0 ... 2 ) --> V )
44		fnfzo0hash 11075	. . . . . . . . 9 |- ( ( 2 e. NN0 /\ F : ( 0..^ 2 ) --> dom I ) -> (♯ ` F ) = 2 )
45	40, 39, 44	sylancr 414	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> (♯ ` F ) = 2 )
46	45	oveq2d 6026	. . . . . . 7 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> ( 0 ... (♯ ` F ) ) = ( 0 ... 2 ) )
47	46	feq2d 5464	. . . . . 6 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> ( P : ( 0 ... (♯ ` F ) ) --> V <-> P : ( 0 ... 2 ) --> V ) )
48	43, 47	mpbird 167	. . . . 5 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> P : ( 0 ... (♯ ` F ) ) --> V )
49		simprl3 1068	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) )
50	45, 27	syl 14	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> ( 0..^ (♯ ` F ) ) = { 0 , 1 } )
51	49, 50	raleqtrrdv 2738	. . . . . . 7 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> A. k e. ( 0..^ (♯ ` F ) )DECID ( P ` k ) = ( P ` ( k + 1 ) ) )
52		simprr 531	. . . . . . . . 9 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
53	52, 35	sylibr 134	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> A. k e. { 0 , 1 } ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
54	53, 50	raleqtrrdv 2738	. . . . . . 7 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
55	51, 54	jca 306	. . . . . 6 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> ( A. k e. ( 0..^ (♯ ` F ) )DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
56		r19.26 2657	. . . . . 6 |- ( A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) <-> ( A. k e. ( 0..^ (♯ ` F ) )DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
57	55, 56	sylibr 134	. . . . 5 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
58	42, 48, 57	3jca 1201	. . . 4 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
59	4	adantr 276	. . . 4 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) )
60	58, 59	mpbird 167	. . 3 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> F (Walks ` G ) P )
61		0z 9473	. . . . . . . . 9 |- 0 e. ZZ
62		2z 9490	. . . . . . . . 9 |- 2 e. ZZ
63		fzofig 10671	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ 2 e. ZZ ) -> ( 0..^ 2 ) e. Fin )
64	61, 62, 63	mp2an 426	. . . . . . . 8 |- ( 0..^ 2 ) e. Fin
65		fex 5875	. . . . . . . 8 |- ( ( F : ( 0..^ 2 ) --> dom I /\ ( 0..^ 2 ) e. Fin ) -> F e. _V )
66	64, 65	mpan2 425	. . . . . . 7 |- ( F : ( 0..^ 2 ) --> dom I -> F e. _V )
67		ffun 5479	. . . . . . 7 |- ( F : ( 0..^ 2 ) --> dom I -> Fun F )
68		fundmeng 6973	. . . . . . 7 |- ( ( F e. _V /\ Fun F ) -> dom F ~~ F )
69	66, 67, 68	syl2anc 411	. . . . . 6 |- ( F : ( 0..^ 2 ) --> dom I -> dom F ~~ F )
70	69	ensymd 6948	. . . . 5 |- ( F : ( 0..^ 2 ) --> dom I -> F ~~ dom F )
71		fdm 5482	. . . . . . 7 |- ( F : ( 0..^ 2 ) --> dom I -> dom F = ( 0..^ 2 ) )
72	71, 26	eqtrdi 2278	. . . . . 6 |- ( F : ( 0..^ 2 ) --> dom I -> dom F = { 0 , 1 } )
73		1z 9488	. . . . . . 7 |- 1 e. ZZ
74		0ne1 9193	. . . . . . 7 |- 0 =/= 1
75		pr2nelem 7380	. . . . . . 7 |- ( ( 0 e. ZZ /\ 1 e. ZZ /\ 0 =/= 1 ) -> { 0 , 1 } ~~ 2o )
76	61, 73, 74, 75	mp3an 1371	. . . . . 6 |- { 0 , 1 } ~~ 2o
77	72, 76	eqbrtrdi 4122	. . . . 5 |- ( F : ( 0..^ 2 ) --> dom I -> dom F ~~ 2o )
78		entr 6949	. . . . 5 |- ( ( F ~~ dom F /\ dom F ~~ 2o ) -> F ~~ 2o )
79	70, 77, 78	syl2anc 411	. . . 4 |- ( F : ( 0..^ 2 ) --> dom I -> F ~~ 2o )
80	39, 79	syl 14	. . 3 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> F ~~ 2o )
81	60, 80	jca 306	. 2 |- ( ( G e. UPGraph /\ ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) -> ( F (Walks ` G ) P /\ F ~~ 2o ) )
82	38, 81	impbida 598	1 |- ( G e. UPGraph -> ( ( F (Walks ` G ) P /\ F ~~ 2o ) <-> ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   _Vcvv 2799   {cpr 3667   class class class wbr 4083   dom cdm 4720   Fun wfun 5315   -->wf 5317   ` cfv 5321  (class class class)co 6010   2oc2o 6567   ~~ cen 6898   Fincfn 6900   0cc0 8015   1c1 8016   + caddc 8018   2c2 9177   NN0cn0 9385   ZZcz 9462   ...cfz 10221  ..^cfzo 10355  ♯chash 11014  Word cword 11089  Vtxcvtx 15834  iEdgciedg 15835  UPGraphcupgr 15912  Walkscwlks 16089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-frec 6548  df-1o 6573  df-2o 6574  df-oadd 6577  df-er 6693  df-map 6810  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-z 9463  df-dec 9595  df-uz 9739  df-fz 10222  df-fzo 10356  df-ihash 11015  df-word 11090  df-ndx 13056  df-slot 13057  df-base 13059  df-edgf 15827  df-vtx 15836  df-iedg 15837  df-edg 15880  df-uhgrm 15890  df-upgren 15914  df-wlks 16090

This theorem is referenced by: (None)