Theorem upgriswlkdc 16301

Description: Properties of a pair of functions to be a walk in a pseudograph. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 28-Oct-2021.)

Hypotheses

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

Assertion

Ref	Expression
upgriswlkdc	|- ( 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 ) ) } ) ) ) )

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

Proof of Theorem upgriswlkdc

Step	Hyp	Ref	Expression
1		upgriswlk.v	. . 3 |- V = (Vtx ` G )
2		upgriswlk.i	. . 3 |- I = (iEdg ` G )
3	1, 2	iswlkg 16270	. 2 |- ( G e. UPGraph -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
4		ifpdc 988	. . . . . . . . 9 |- (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> DECID ( P ` k ) = ( P ` ( k + 1 ) ) )
5	4	adantl 277	. . . . . . . 8 |- ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> DECID ( P ` k ) = ( P ` ( k + 1 ) ) )
6		df-ifp 987	. . . . . . . . . 10 |- (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) <-> ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) } ) \/ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
7		dfsn2 3687	. . . . . . . . . . . . . . . 16 |- { ( P ` k ) } = { ( P ` k ) , ( P ` k ) }
8		preq2 3753	. . . . . . . . . . . . . . . 16 |- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> { ( P ` k ) , ( P ` k ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
9	7, 8	eqtrid 2276	. . . . . . . . . . . . . . 15 |- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> { ( P ` k ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
10	9	eqeq2d 2243	. . . . . . . . . . . . . 14 |- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> ( ( I ` ( F ` k ) ) = { ( P ` k ) } <-> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
11	10	biimpa 296	. . . . . . . . . . . . 13 |- ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) } ) -> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
12	11	a1d 22	. . . . . . . . . . . 12 |- ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) } ) -> ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
13		eqid 2231	. . . . . . . . . . . . . . . . 17 |- (Edg ` G ) = (Edg ` G )
14	2, 13	upgredginwlk 16297	. . . . . . . . . . . . . . . 16 |- ( ( G e. UPGraph /\ F e. Word dom I ) -> ( k e. ( 0..^ (♯ ` F ) ) -> ( I ` ( F ` k ) ) e. (Edg ` G ) ) )
15	14	adantrr 479	. . . . . . . . . . . . . . 15 |- ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) -> ( k e. ( 0..^ (♯ ` F ) ) -> ( I ` ( F ` k ) ) e. (Edg ` G ) ) )
16	15	imp 124	. . . . . . . . . . . . . 14 |- ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( I ` ( F ` k ) ) e. (Edg ` G ) )
17		simp-4l 543	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> G e. UPGraph )
18		simplr 529	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( I ` ( F ` k ) ) e. (Edg ` G ) )
19		simprr 533	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
20		simprr 533	. . . . . . . . . . . . . . . . . . . 20 |- ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) -> P : ( 0 ... (♯ ` F ) ) --> V )
21	20	ad5ant12 518	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> P : ( 0 ... (♯ ` F ) ) --> V )
22		elfzofz 10460	. . . . . . . . . . . . . . . . . . . 20 |- ( k e. ( 0..^ (♯ ` F ) ) -> k e. ( 0 ... (♯ ` F ) ) )
23	22	ad3antlr 493	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> k e. ( 0 ... (♯ ` F ) ) )
24	21, 23	ffvelcdmd 5791	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( P ` k ) e. V )
25	24	elexd 2817	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( P ` k ) e. _V )
26		fzofzp1 10535	. . . . . . . . . . . . . . . . . . . 20 |- ( k e. ( 0..^ (♯ ` F ) ) -> ( k + 1 ) e. ( 0 ... (♯ ` F ) ) )
27	26	ad3antlr 493	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( k + 1 ) e. ( 0 ... (♯ ` F ) ) )
28	21, 27	ffvelcdmd 5791	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( P ` ( k + 1 ) ) e. V )
29	28	elexd 2817	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( P ` ( k + 1 ) ) e. _V )
30		neqne 2411	. . . . . . . . . . . . . . . . . 18 |- ( -. ( P ` k ) = ( P ` ( k + 1 ) ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) )
31	30	ad2antrl 490	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) )
32	1, 13	upgredgpr 16090	. . . . . . . . . . . . . . . . 17 |- ( ( ( G e. UPGraph /\ ( I ` ( F ` k ) ) e. (Edg ` G ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) /\ ( ( P ` k ) e. _V /\ ( P ` ( k + 1 ) ) e. _V /\ ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = ( I ` ( F ` k ) ) )
33	17, 18, 19, 25, 29, 31, 32	syl33anc 1289	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = ( I ` ( F ` k ) ) )
34	33	eqcomd 2237	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ ( I ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
35	34	exp31 364	. . . . . . . . . . . . . 14 |- ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( ( I ` ( F ` k ) ) e. (Edg ` G ) -> ( ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
36	16, 35	mpd 13	. . . . . . . . . . . . 13 |- ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
37	36	com12 30	. . . . . . . . . . . 12 |- ( ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
38	12, 37	jaoi 724	. . . . . . . . . . 11 |- ( ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) } ) \/ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
39	38	com12 30	. . . . . . . . . 10 |- ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) } ) \/ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
40	6, 39	biimtrid 152	. . . . . . . . 9 |- ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
41	40	imp 124	. . . . . . . 8 |- ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
42	5, 41	jca 306	. . . . . . 7 |- ( ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
43	42	ex 115	. . . . . 6 |- ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
44		ifpprsnssdc 3783	. . . . . . 7 |- ( ( ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } /\ DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) -> if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
45	44	ancoms 268	. . . . . 6 |- ( (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
46	43, 45	impbid1 142	. . . . 5 |- ( ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) <-> (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
47	46	ralbidva 2529	. . . 4 |- ( ( G e. UPGraph /\ ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) ) -> ( A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) <-> A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
48	47	pm5.32da 452	. . 3 |- ( G e. UPGraph -> ( ( ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) <-> ( ( 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 ) ) } ) ) ) )
49		df-3an 1007	. . 3 |- ( ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) <-> ( ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
50		df-3an 1007	. . 3 |- ( ( 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 ) ) } ) ) <-> ( ( 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 ) ) } ) ) )
51	48, 49, 50	3bitr4g 223	. 2 |- ( G e. UPGraph -> ( ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) <-> ( 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 ) ) } ) ) ) )
52	3, 51	bitrd 188	1 |- ( 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 ) ) } ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842  if-wif 986   /\ w3a 1005   = wceq 1398   e. wcel 2202   =/= wne 2403   A.wral 2511   _Vcvv 2803   C_ wss 3201   {csn 3673   {cpr 3674   class class class wbr 4093   dom cdm 4731   -->wf 5329   ` cfv 5333  (class class class)co 6028   0cc0 8092   1c1 8093   + caddc 8095   ...cfz 10305  ..^cfzo 10439  ♯chash 11100  Word cword 11179  Vtxcvtx 15953  iEdgciedg 15954  Edgcedg 15998  UPGraphcupgr 16032  Walkscwlks 16258

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 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208

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-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-9 9268  df-n0 9462  df-z 9541  df-dec 9673  df-uz 9817  df-fz 10306  df-fzo 10440  df-ihash 11101  df-word 11180  df-ndx 13165  df-slot 13166  df-base 13168  df-edgf 15946  df-vtx 15955  df-iedg 15956  df-edg 15999  df-uhgrm 16010  df-upgren 16034  df-wlks 16259

This theorem is referenced by:  upgrwlkedg  16302  upgrwlkcompim  16303  upgrwlkvtxedg  16305  upgr2wlkdc  16318