Theorem upgr2wlkdc 16231

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 531	. . . . . . . 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 16214	. . . . . . . . 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 1035	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> F e. Word dom I )
8		wrdf 11120	. . . . . 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 533	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> F ~~ 2o )
11		hash2en 11108	. . . . . . . . 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 6034	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( 0..^ (♯ ` F ) ) = ( 0..^ 2 ) )
15	14	feq2d 5470	. . . . 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 1036	. . . . 5 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> P : ( 0 ... (♯ ` F ) ) --> V )
18	13	oveq2d 6034	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( 0 ... (♯ ` F ) ) = ( 0 ... 2 ) )
19	18	feq2d 5470	. . . . 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 1037	. . . . . 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 2595	. . . . . 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 6026	. . . . . . 7 |- ( (♯ ` F ) = 2 -> ( 0..^ (♯ ` F ) ) = ( 0..^ 2 ) )
26		fzo0to2pr 10464	. . . . . . 7 |- ( 0..^ 2 ) = { 0 , 1 }
27	25, 26	eqtrdi 2280	. . . . . 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 2738	. . . 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 1203	. . 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 2595	. . . . 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 2736	. . . . 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 16230	. . . . 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 1068	. . . . . 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 9419	. . . . . 6 |- 2 e. NN0
41		iswrdinn0 11119	. . . . . 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 1069	. . . . . 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 11100	. . . . . . . . 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 6034	. . . . . . 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 5470	. . . . . 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 1070	. . . . . . . 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 2740	. . . . . . 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 533	. . . . . . . . 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 2740	. . . . . . 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 2659	. . . . . 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 1203	. . . 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 9490	. . . . . . . . 9 |- 0 e. ZZ
62		2z 9507	. . . . . . . . 9 |- 2 e. ZZ
63		fzofig 10695	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ 2 e. ZZ ) -> ( 0..^ 2 ) e. Fin )
64	61, 62, 63	mp2an 426	. . . . . . . 8 |- ( 0..^ 2 ) e. Fin
65		fex 5883	. . . . . . . 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 5485	. . . . . . 7 |- ( F : ( 0..^ 2 ) --> dom I -> Fun F )
68		fundmeng 6982	. . . . . . 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 6957	. . . . 5 |- ( F : ( 0..^ 2 ) --> dom I -> F ~~ dom F )
71		fdm 5488	. . . . . . 7 |- ( F : ( 0..^ 2 ) --> dom I -> dom F = ( 0..^ 2 ) )
72	71, 26	eqtrdi 2280	. . . . . 6 |- ( F : ( 0..^ 2 ) --> dom I -> dom F = { 0 , 1 } )
73		1z 9505	. . . . . . 7 |- 1 e. ZZ
74		0ne1 9210	. . . . . . 7 |- 0 =/= 1
75		pr2nelem 7396	. . . . . . 7 |- ( ( 0 e. ZZ /\ 1 e. ZZ /\ 0 =/= 1 ) -> { 0 , 1 } ~~ 2o )
76	61, 73, 74, 75	mp3an 1373	. . . . . 6 |- { 0 , 1 } ~~ 2o
77	72, 76	eqbrtrdi 4127	. . . . 5 |- ( F : ( 0..^ 2 ) --> dom I -> dom F ~~ 2o )
78		entr 6958	. . . . 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 600	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 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   A.wral 2510   _Vcvv 2802   {cpr 3670   class class class wbr 4088   dom cdm 4725   Fun wfun 5320   -->wf 5322   ` cfv 5326  (class class class)co 6018   2oc2o 6576   ~~ cen 6907   Fincfn 6909   0cc0 8032   1c1 8033   + caddc 8035   2c2 9194   NN0cn0 9402   ZZcz 9479   ...cfz 10243  ..^cfzo 10377  ♯chash 11038  Word cword 11114  Vtxcvtx 15866  iEdgciedg 15867  UPGraphcupgr 15945  Walkscwlks 16171

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-ifp 986  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-map 6819  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11115  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-edg 15912  df-uhgrm 15923  df-upgren 15947  df-wlks 16172

This theorem is referenced by: (None)