Theorem upgriswlkdc 16132

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 16101	. 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 985	. . . . . . . . 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 984	. . . . . . . . . 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 3680	. . . . . . . . . . . . . . . 16 |- { ( P ` k ) } = { ( P ` k ) , ( P ` k ) }
8		preq2 3744	. . . . . . . . . . . . . . . 16 |- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> { ( P ` k ) , ( P ` k ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
9	7, 8	eqtrid 2274	. . . . . . . . . . . . . . 15 |- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> { ( P ` k ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
10	9	eqeq2d 2241	. . . . . . . . . . . . . 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 2229	. . . . . . . . . . . . . . . . 17 |- (Edg ` G ) = (Edg ` G )
14	2, 13	upgredginwlk 16128	. . . . . . . . . . . . . . . 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 541	. . . . . . . . . . . . . . . . 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 528	. . . . . . . . . . . . . . . . 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 531	. . . . . . . . . . . . . . . . 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 531	. . . . . . . . . . . . . . . . . . . 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 10376	. . . . . . . . . . . . . . . . . . . 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 5776	. . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . 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 10450	. . . . . . . . . . . . . . . . . . . 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 5776	. . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . 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 2408	. . . . . . . . . . . . . . . . . 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 15968	. . . . . . . . . . . . . . . . 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 1286	. . . . . . . . . . . . . . . 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 2235	. . . . . . . . . . . . . . 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 721	. . . . . . . . . . 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 3774	. . . . . . 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 2526	. . . 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 1004	. . 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 1004	. . 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 713  DECID wdc 839  if-wif 983   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   _Vcvv 2799   C_ wss 3197   {csn 3666   {cpr 3667   class class class wbr 4083   dom cdm 4720   -->wf 5317   ` cfv 5321  (class class class)co 6010   0cc0 8015   1c1 8016   + caddc 8018   ...cfz 10221  ..^cfzo 10355  ♯chash 11014  Word cword 11089  Vtxcvtx 15834  iEdgciedg 15835  Edgcedg 15879  UPGraphcupgr 15912  Walkscwlks 16089

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-1o 6573  df-2o 6574  df-er 6693  df-map 6810  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-z 9463  df-dec 9595  df-uz 9739  df-fz 10222  df-fzo 10356  df-ihash 11015  df-word 11090  df-ndx 13056  df-slot 13057  df-base 13059  df-edgf 15827  df-vtx 15836  df-iedg 15837  df-edg 15880  df-uhgrm 15890  df-upgren 15914  df-wlks 16090

This theorem is referenced by:  upgrwlkedg  16133  upgrwlkcompim  16134  upgrwlkvtxedg  16136  upgr2wlkdc  16147