Theorem upgr2wlkdc 16364

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 16347	. . . . . . . . 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 1036	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> F e. Word dom I )
8		wrdf 11226	. . . . . 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 11211	. . . . . . . . 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 6065	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( 0..^ (♯ ` F ) ) = ( 0..^ 2 ) )
15	14	feq2d 5495	. . . . 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 1037	. . . . 5 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> P : ( 0 ... (♯ ` F ) ) --> V )
18	13	oveq2d 6065	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( 0 ... (♯ ` F ) ) = ( 0 ... 2 ) )
19	18	feq2d 5495	. . . . 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 1038	. . . . . 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 2605	. . . . . 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 6057	. . . . . . 7 |- ( (♯ ` F ) = 2 -> ( 0..^ (♯ ` F ) ) = ( 0..^ 2 ) )
26		fzo0to2pr 10562	. . . . . . 7 |- ( 0..^ 2 ) = { 0 , 1 }
27	25, 26	eqtrdi 2281	. . . . . 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 2748	. . . 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 1204	. . 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 2605	. . . . 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 2746	. . . . 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 16363	. . . . 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 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 ) } ) ) ) -> F : ( 0..^ 2 ) --> dom I )
40		2nn0 9512	. . . . . 6 |- 2 e. NN0
41		iswrdinn0 11225	. . . . . 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 1070	. . . . . 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 11198	. . . . . . . . 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 6065	. . . . . . 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 5495	. . . . . 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 1071	. . . . . . . 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 2750	. . . . . . 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 2750	. . . . . . 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 2669	. . . . . 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 1204	. . . 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 9587	. . . . . . . . 9 |- 0 e. ZZ
62		2z 9604	. . . . . . . . 9 |- 2 e. ZZ
63		fzofig 10793	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ 2 e. ZZ ) -> ( 0..^ 2 ) e. Fin )
64	61, 62, 63	mp2an 426	. . . . . . . 8 |- ( 0..^ 2 ) e. Fin
65		fex 5914	. . . . . . . 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 5510	. . . . . . 7 |- ( F : ( 0..^ 2 ) --> dom I -> Fun F )
68		fundmeng 7047	. . . . . . 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 7022	. . . . 5 |- ( F : ( 0..^ 2 ) --> dom I -> F ~~ dom F )
71		fdm 5513	. . . . . . 7 |- ( F : ( 0..^ 2 ) --> dom I -> dom F = ( 0..^ 2 ) )
72	71, 26	eqtrdi 2281	. . . . . 6 |- ( F : ( 0..^ 2 ) --> dom I -> dom F = { 0 , 1 } )
73		1z 9602	. . . . . . 7 |- 1 e. ZZ
74		0ne1 9303	. . . . . . 7 |- 0 =/= 1
75		pr2nelem 7487	. . . . . . 7 |- ( ( 0 e. ZZ /\ 1 e. ZZ /\ 0 =/= 1 ) -> { 0 , 1 } ~~ 2o )
76	61, 73, 74, 75	mp3an 1374	. . . . . 6 |- { 0 , 1 } ~~ 2o
77	72, 76	eqbrtrdi 4147	. . . . 5 |- ( F : ( 0..^ 2 ) --> dom I -> dom F ~~ 2o )
78		entr 7023	. . . . 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 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   A.wral 2520   _Vcvv 2812   {cpr 3689   class class class wbr 4108   dom cdm 4748   Fun wfun 5345   -->wf 5347   ` cfv 5351  (class class class)co 6049   2oc2o 6640   ~~ cen 6972   Fincfn 6974   0cc0 8126   1c1 8127   + caddc 8129   2c2 9287   NN0cn0 9495   ZZcz 9576   ...cfz 10341  ..^cfzo 10475  ♯chash 11136  Word cword 11220  Vtxcvtx 15999  iEdgciedg 16000  UPGraphcupgr 16078  Walkscwlks 16304

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-map 6883  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300  df-8 9301  df-9 9302  df-n0 9496  df-z 9577  df-dec 9709  df-uz 9853  df-fz 10342  df-fzo 10476  df-ihash 11137  df-word 11221  df-ndx 13207  df-slot 13208  df-base 13210  df-edgf 15992  df-vtx 16001  df-iedg 16002  df-edg 16045  df-uhgrm 16056  df-upgren 16080  df-wlks 16305

This theorem is referenced by: (None)