Theorem upgr2wlkdc 16630

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 535	. . . . . . . 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 16613	. . . . . . . . 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 1040	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> F e. Word dom I )
8		wrdf 11312	. . . . . 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 537	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> F ~~ 2o )
11		hash2en 11297	. . . . . . . . 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 6101	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( 0..^ (♯ ` F ) ) = ( 0..^ 2 ) )
15	14	feq2d 5521	. . . . 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 1041	. . . . 5 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> P : ( 0 ... (♯ ` F ) ) --> V )
18	13	oveq2d 6101	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( 0 ... (♯ ` F ) ) = ( 0 ... 2 ) )
19	18	feq2d 5521	. . . . 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 1042	. . . . . 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 2613	. . . . . 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 6093	. . . . . . 7 |- ( (♯ ` F ) = 2 -> ( 0..^ (♯ ` F ) ) = ( 0..^ 2 ) )
26		fzo0to2pr 10638	. . . . . . 7 |- ( 0..^ 2 ) = { 0 , 1 }
27	25, 26	eqtrdi 2287	. . . . . 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 2757	. . . 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 1208	. . 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 2613	. . . . 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 2755	. . . . 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 16629	. . . . 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 1073	. . . . . 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 9582	. . . . . 6 |- 2 e. NN0
41		iswrdinn0 11311	. . . . . 6 |- ( ( F : ( 0..^ 2 ) --> dom I /\ 2 e. NN0 ) -> F e. Word dom I )
42	39, 40, 41	sylancl 417	. . . . 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 1074	. . . . . 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 11280	. . . . . . . . 9 |- ( ( 2 e. NN0 /\ F : ( 0..^ 2 ) --> dom I ) -> (♯ ` F ) = 2 )
45	40, 39, 44	sylancr 418	. . . . . . . 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 6101	. . . . . . 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 5521	. . . . . 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 1075	. . . . . . . 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 2759	. . . . . . 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 537	. . . . . . . . 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 2759	. . . . . . 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 2677	. . . . . 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 1208	. . . 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 9657	. . . . . . . . 9 |- 0 e. ZZ
62		2z 9674	. . . . . . . . 9 |- 2 e. ZZ
63		fzofig 10871	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ 2 e. ZZ ) -> ( 0..^ 2 ) e. Fin )
64	61, 62, 63	mp2an 430	. . . . . . . 8 |- ( 0..^ 2 ) e. Fin
65		fex 5947	. . . . . . . 8 |- ( ( F : ( 0..^ 2 ) --> dom I /\ ( 0..^ 2 ) e. Fin ) -> F e. _V )
66	64, 65	mpan2 429	. . . . . . 7 |- ( F : ( 0..^ 2 ) --> dom I -> F e. _V )
67		ffun 5536	. . . . . . 7 |- ( F : ( 0..^ 2 ) --> dom I -> Fun F )
68		fundmeng 7095	. . . . . . 7 |- ( ( F e. _V /\ Fun F ) -> dom F ~~ F )
69	66, 67, 68	syl2anc 415	. . . . . 6 |- ( F : ( 0..^ 2 ) --> dom I -> dom F ~~ F )
70	69	ensymd 7070	. . . . 5 |- ( F : ( 0..^ 2 ) --> dom I -> F ~~ dom F )
71		fdm 5539	. . . . . . 7 |- ( F : ( 0..^ 2 ) --> dom I -> dom F = ( 0..^ 2 ) )
72	71, 26	eqtrdi 2287	. . . . . 6 |- ( F : ( 0..^ 2 ) --> dom I -> dom F = { 0 , 1 } )
73		1z 9672	. . . . . . 7 |- 1 e. ZZ
74		0ne1 9372	. . . . . . 7 |- 0 =/= 1
75		pr2nelem 7537	. . . . . . 7 |- ( ( 0 e. ZZ /\ 1 e. ZZ /\ 0 =/= 1 ) -> { 0 , 1 } ~~ 2o )
76	61, 73, 74, 75	mp3an 1378	. . . . . 6 |- { 0 , 1 } ~~ 2o
77	72, 76	eqbrtrdi 4169	. . . . 5 |- ( F : ( 0..^ 2 ) --> dom I -> dom F ~~ 2o )
78		entr 7071	. . . . 5 |- ( ( F ~~ dom F /\ dom F ~~ 2o ) -> F ~~ 2o )
79	70, 77, 78	syl2anc 415	. . . 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 604	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 ) } ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   _Vcvv 2821   {cpr 3710   class class class wbr 4130   dom cdm 4774   Fun wfun 5371   -->wf 5373   ` cfv 5377  (class class class)co 6085   2oc2o 6681   ~~ cen 7020   Fincfn 7022   0cc0 8179   1c1 8180   + caddc 8182   2c2 9356   NN0cn0 9565   ZZcz 9646   ...cfz 10413  ..^cfzo 10551  ♯chash 11216  Word cword 11306  Vtxcvtx 16265  iEdgciedg 16266  UPGraphcupgr 16344  Walkscwlks 16570

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-z 9647  df-dec 9780  df-uz 9924  df-fz 10414  df-fzo 10552  df-ihash 11217  df-word 11307  df-ndx 13357  df-slot 13358  df-base 13360  df-edgf 16258  df-vtx 16267  df-iedg 16268  df-edg 16311  df-uhgrm 16322  df-upgren 16346  df-wlks 16571

This theorem is used by: (None)