Theorem wlkiswwlks2 26761

Description: A walk as word corresponds to the sequence of vertices in a walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)

Assertion

Ref	Expression
wlkiswwlks2	|- ( G e. USPGraph -> ( P e. (WWalks ` G ) -> E. f f (Walks ` G ) P ) )

Distinct variable groups:   f, G   P, f

Proof of Theorem wlkiswwlks2

Dummy variables i x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
2	1	wwlkbp 26732	. . 3 |- ( P e. (WWalks ` G ) -> ( G e. _V /\ P e. Word (Vtx ` G ) ) )
3		eqid 2622	. . . . 5 |- (Edg ` G ) = (Edg ` G )
4	1, 3	iswwlks 26728	. . . 4 |- ( P e. (WWalks ` G ) <-> ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) )
5		ovex 6678	. . . . . . . . . . . . . . 15 |- ( 0..^ ( ( # ` P ) - 1 ) ) e. _V
6		mptexg 6484	. . . . . . . . . . . . . . 15 |- ( ( 0..^ ( ( # ` P ) - 1 ) ) e. _V -> ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) e. _V )
7	5, 6	mp1i 13	. . . . . . . . . . . . . 14 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) -> ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) e. _V )
8		simprr 796	. . . . . . . . . . . . . . . . . 18 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) -> G e. USPGraph )
9		simplr 792	. . . . . . . . . . . . . . . . . 18 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) -> P e. Word (Vtx ` G ) )
10		hashge1 13178	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P e. Word (Vtx ` G ) /\ P =/= (/) ) -> 1 <_ ( # ` P ) )
11	10	ancoms 469	. . . . . . . . . . . . . . . . . . 19 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) -> 1 <_ ( # ` P ) )
12	11	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) -> 1 <_ ( # ` P ) )
13	8, 9, 12	3jca 1242	. . . . . . . . . . . . . . . . 17 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) -> ( G e. USPGraph /\ P e. Word (Vtx ` G ) /\ 1 <_ ( # ` P ) ) )
14	13	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) /\ f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) -> ( G e. USPGraph /\ P e. Word (Vtx ` G ) /\ 1 <_ ( # ` P ) ) )
15		edgval 25941	. . . . . . . . . . . . . . . . . . . 20 |- (Edg ` G ) = ran (iEdg ` G )
16	15	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) /\ f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) -> (Edg ` G ) = ran (iEdg ` G ) )
17	16	eleq2d 2687	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) /\ f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran (iEdg ` G ) ) )
18	17	ralbidv 2986	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) /\ f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran (iEdg ` G ) ) )
19	18	biimpd 219	. . . . . . . . . . . . . . . 16 |- ( ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) /\ f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran (iEdg ` G ) ) )
20		eqid 2622	. . . . . . . . . . . . . . . . 17 |- ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) )
21		eqid 2622	. . . . . . . . . . . . . . . . 17 |- (iEdg ` G ) = (iEdg ` G )
22	20, 21	wlkiswwlks2lem6 26760	. . . . . . . . . . . . . . . 16 |- ( ( G e. USPGraph /\ P e. Word (Vtx ` G ) /\ 1 <_ ( # ` P ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran (iEdg ` G ) -> ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) ( (iEdg ` G ) ` ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
23	14, 19, 22	sylsyld 61	. . . . . . . . . . . . . . 15 |- ( ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) /\ f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) ( (iEdg ` G ) ` ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
24		eleq1 2689	. . . . . . . . . . . . . . . . . 18 |- ( f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) -> ( f e. Word dom (iEdg ` G ) <-> ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) e. Word dom (iEdg ` G ) ) )
25		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) -> ( # ` f ) = ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) )
26	25	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 |- ( f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) -> ( 0 ... ( # ` f ) ) = ( 0 ... ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) )
27	26	feq2d 6031	. . . . . . . . . . . . . . . . . 18 |- ( f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) -> ( P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) <-> P : ( 0 ... ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) --> (Vtx ` G ) ) )
28	25	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 |- ( f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) -> ( 0..^ ( # ` f ) ) = ( 0..^ ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) )
29		fveq1 6190	. . . . . . . . . . . . . . . . . . . . 21 |- ( f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) -> ( f ` i ) = ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ` i ) )
30	29	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 |- ( f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) -> ( (iEdg ` G ) ` ( f ` i ) ) = ( (iEdg ` G ) ` ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ` i ) ) )
31	30	eqeq1d 2624	. . . . . . . . . . . . . . . . . . 19 |- ( f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) -> ( ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( (iEdg ` G ) ` ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
32	28, 31	raleqbidv 3152	. . . . . . . . . . . . . . . . . 18 |- ( f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) -> ( A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> A. i e. ( 0..^ ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) ( (iEdg ` G ) ` ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
33	24, 27, 32	3anbi123d 1399	. . . . . . . . . . . . . . . . 17 |- ( f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) -> ( ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) <-> ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) ( (iEdg ` G ) ` ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
34	33	imbi2d 330	. . . . . . . . . . . . . . . 16 |- ( f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) -> ( ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) <-> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) ( (iEdg ` G ) ` ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) )
35	34	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) /\ f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) -> ( ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) <-> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) ) ( (iEdg ` G ) ` ( ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) )
36	23, 35	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) /\ f = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' (iEdg ` G ) ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
37	7, 36	spcimedv 3292	. . . . . . . . . . . . 13 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> E. f ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
38	37	ex 450	. . . . . . . . . . . 12 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) -> ( ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> E. f ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) )
39	38	com23 86	. . . . . . . . . . 11 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> ( ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) -> E. f ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) )
40	39	3impia 1261	. . . . . . . . . 10 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ G e. USPGraph ) -> E. f ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
41	40	expd 452	. . . . . . . . 9 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( ( G e. _V /\ P e. Word (Vtx ` G ) ) -> ( G e. USPGraph -> E. f ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) )
42	41	impcom 446	. . . . . . . 8 |- ( ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> ( G e. USPGraph -> E. f ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
43	42	imp 445	. . . . . . 7 |- ( ( ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ G e. USPGraph ) -> E. f ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
44		uspgrupgr 26071	. . . . . . . . . 10 |- ( G e. USPGraph -> G e. UPGraph )
45	1, 21	upgriswlk 26537	. . . . . . . . . 10 |- ( G e. UPGraph -> ( f (Walks ` G ) P <-> ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
46	44, 45	syl 17	. . . . . . . . 9 |- ( G e. USPGraph -> ( f (Walks ` G ) P <-> ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
47	46	adantl 482	. . . . . . . 8 |- ( ( ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ G e. USPGraph ) -> ( f (Walks ` G ) P <-> ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
48	47	exbidv 1850	. . . . . . 7 |- ( ( ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ G e. USPGraph ) -> ( E. f f (Walks ` G ) P <-> E. f ( f e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
49	43, 48	mpbird 247	. . . . . 6 |- ( ( ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ G e. USPGraph ) -> E. f f (Walks ` G ) P )
50	49	ex 450	. . . . 5 |- ( ( ( G e. _V /\ P e. Word (Vtx ` G ) ) /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> ( G e. USPGraph -> E. f f (Walks ` G ) P ) )
51	50	ex 450	. . . 4 |- ( ( G e. _V /\ P e. Word (Vtx ` G ) ) -> ( ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( G e. USPGraph -> E. f f (Walks ` G ) P ) ) )
52	4, 51	syl5bi 232	. . 3 |- ( ( G e. _V /\ P e. Word (Vtx ` G ) ) -> ( P e. (WWalks ` G ) -> ( G e. USPGraph -> E. f f (Walks ` G ) P ) ) )
53	2, 52	mpcom 38	. 2 |- ( P e. (WWalks ` G ) -> ( G e. USPGraph -> E. f f (Walks ` G ) P ) )
54	53	com12 32	1 |- ( G e. USPGraph -> ( P e. (WWalks ` G ) -> E. f f (Walks ` G ) P ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   (/)c0 3915   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UPGraph cupgr 25975   USPGraph cuspgr 26043  Walkscwlks 26492  WWalkscwwlks 26717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-wlks 26495  df-wwlks 26722

This theorem is referenced by:  wlkiswwlks  26762  wlklnwwlkln2  26769