Theorem upgr2wlkdc 16295

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 16278	. . . . . . . . 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 11166	. . . . . 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 11151	. . . . . . . . 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 6044	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( 0..^ (♯ ` F ) ) = ( 0..^ 2 ) )
15	14	feq2d 5477	. . . . 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 6044	. . . . . 6 |- ( ( G e. UPGraph /\ ( F (Walks ` G ) P /\ F ~~ 2o ) ) -> ( 0 ... (♯ ` F ) ) = ( 0 ... 2 ) )
19	18	feq2d 5477	. . . . 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 2596	. . . . . 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 6036	. . . . . . 7 |- ( (♯ ` F ) = 2 -> ( 0..^ (♯ ` F ) ) = ( 0..^ 2 ) )
26		fzo0to2pr 10507	. . . . . . 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 2739	. . . 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 2596	. . . . 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 2737	. . . . 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 16294	. . . . 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 9462	. . . . . 6 |- 2 e. NN0
41		iswrdinn0 11165	. . . . . 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 11143	. . . . . . . . 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 6044	. . . . . . 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 5477	. . . . . 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 2741	. . . . . . 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 2741	. . . . . . 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 2660	. . . . . 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 9533	. . . . . . . . 9 |- 0 e. ZZ
62		2z 9550	. . . . . . . . 9 |- 2 e. ZZ
63		fzofig 10738	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ 2 e. ZZ ) -> ( 0..^ 2 ) e. Fin )
64	61, 62, 63	mp2an 426	. . . . . . . 8 |- ( 0..^ 2 ) e. Fin
65		fex 5893	. . . . . . . 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 5492	. . . . . . 7 |- ( F : ( 0..^ 2 ) --> dom I -> Fun F )
68		fundmeng 7025	. . . . . . 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 7000	. . . . 5 |- ( F : ( 0..^ 2 ) --> dom I -> F ~~ dom F )
71		fdm 5495	. . . . . . 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 9548	. . . . . . 7 |- 1 e. ZZ
74		0ne1 9253	. . . . . . 7 |- 0 =/= 1
75		pr2nelem 7439	. . . . . . 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 4132	. . . . 5 |- ( F : ( 0..^ 2 ) --> dom I -> dom F ~~ 2o )
78		entr 7001	. . . . 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 2202   =/= wne 2403   A.wral 2511   _Vcvv 2803   {cpr 3674   class class class wbr 4093   dom cdm 4731   Fun wfun 5327   -->wf 5329   ` cfv 5333  (class class class)co 6028   2oc2o 6619   ~~ cen 6950   Fincfn 6952   0cc0 8075   1c1 8076   + caddc 8078   2c2 9237   NN0cn0 9445   ZZcz 9522   ...cfz 10286  ..^cfzo 10420  ♯chash 11081  Word cword 11160  Vtxcvtx 15930  iEdgciedg 15931  UPGraphcupgr 16009  Walkscwlks 16235

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-2o 6626  df-oadd 6629  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9523  df-dec 9655  df-uz 9799  df-fz 10287  df-fzo 10421  df-ihash 11082  df-word 11161  df-ndx 13146  df-slot 13147  df-base 13149  df-edgf 15923  df-vtx 15932  df-iedg 15933  df-edg 15976  df-uhgrm 15987  df-upgren 16011  df-wlks 16236

This theorem is referenced by: (None)