Theorem wlkiswwlksupgr2 26763

Description: A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of wlkiswwlks2 26761 does not require G to be a simple pseudograph, but it requires the Axiom of Choice (ac6 9302) for its proof. Notice that only the existence of a function f can be proven, but, in general, it cannot be "constructed" (as in wlkiswwlks2 26761). (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)

Assertion

Ref	Expression
wlkiswwlksupgr2	|- ( G e. UPGraph -> ( P e. (WWalks ` G ) -> E. f f (Walks ` G ) P ) )

Distinct variable groups:   f, G   P, f

Proof of Theorem wlkiswwlksupgr2

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2622	. . 3 |- (Edg ` G ) = (Edg ` G )
3	1, 2	iswwlks 26728	. 2 |- ( P e. (WWalks ` G ) <-> ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) )
4		edgval 25941	. . . . . . . . . . . . 13 |- (Edg ` G ) = ran (iEdg ` G )
5	4	eleq2i 2693	. . . . . . . . . . . 12 |- ( { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran (iEdg ` G ) )
6		upgruhgr 25997	. . . . . . . . . . . . . . 15 |- ( G e. UPGraph -> G e. UHGraph )
7		eqid 2622	. . . . . . . . . . . . . . . 16 |- (iEdg ` G ) = (iEdg ` G )
8	7	uhgrfun 25961	. . . . . . . . . . . . . . 15 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
9	6, 8	syl 17	. . . . . . . . . . . . . 14 |- ( G e. UPGraph -> Fun (iEdg ` G ) )
10	9	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ G e. UPGraph ) -> Fun (iEdg ` G ) )
11		elrnrexdm 6363	. . . . . . . . . . . . . 14 |- ( Fun (iEdg ` G ) -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran (iEdg ` G ) -> E. x e. dom (iEdg ` G ) { ( P ` i ) , ( P ` ( i + 1 ) ) } = ( (iEdg ` G ) ` x ) ) )
12		eqcom 2629	. . . . . . . . . . . . . . 15 |- ( ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } = ( (iEdg ` G ) ` x ) )
13	12	rexbii 3041	. . . . . . . . . . . . . 14 |- ( E. x e. dom (iEdg ` G ) ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> E. x e. dom (iEdg ` G ) { ( P ` i ) , ( P ` ( i + 1 ) ) } = ( (iEdg ` G ) ` x ) )
14	11, 13	syl6ibr 242	. . . . . . . . . . . . 13 |- ( Fun (iEdg ` G ) -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran (iEdg ` G ) -> E. x e. dom (iEdg ` G ) ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
15	10, 14	syl 17	. . . . . . . . . . . 12 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ G e. UPGraph ) -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran (iEdg ` G ) -> E. x e. dom (iEdg ` G ) ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
16	5, 15	syl5bi 232	. . . . . . . . . . 11 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ G e. UPGraph ) -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> E. x e. dom (iEdg ` G ) ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
17	16	ralimdv 2963	. . . . . . . . . 10 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ G e. UPGraph ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) E. x e. dom (iEdg ` G ) ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
18	17	ex 450	. . . . . . . . 9 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) -> ( G e. UPGraph -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) E. x e. dom (iEdg ` G ) ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
19	18	com23 86	. . . . . . . 8 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) -> ( G e. UPGraph -> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) E. x e. dom (iEdg ` G ) ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
20	19	3impia 1261	. . . . . . 7 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( G e. UPGraph -> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) E. x e. dom (iEdg ` G ) ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
21	20	impcom 446	. . . . . 6 |- ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) E. x e. dom (iEdg ` G ) ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
22		ovex 6678	. . . . . . 7 |- ( 0..^ ( ( # ` P ) - 1 ) ) e. _V
23		fvex 6201	. . . . . . . 8 |- (iEdg ` G ) e. _V
24	23	dmex 7099	. . . . . . 7 |- dom (iEdg ` G ) e. _V
25		fveq2 6191	. . . . . . . 8 |- ( x = ( f ` i ) -> ( (iEdg ` G ) ` x ) = ( (iEdg ` G ) ` ( f ` i ) ) )
26	25	eqeq1d 2624	. . . . . . 7 |- ( x = ( f ` i ) -> ( ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
27	22, 24, 26	ac6 9302	. . . . . 6 |- ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) E. x e. dom (iEdg ` G ) ( (iEdg ` G ) ` x ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } -> E. f ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
28	21, 27	syl 17	. . . . 5 |- ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> E. f ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
29		iswrdi 13309	. . . . . . . . . 10 |- ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) -> f e. Word dom (iEdg ` G ) )
30	29	adantr 481	. . . . . . . . 9 |- ( ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> f e. Word dom (iEdg ` G ) )
31	30	adantl 482	. . . . . . . 8 |- ( ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> f e. Word dom (iEdg ` G ) )
32		wrdfin 13323	. . . . . . . . . . . . . . . 16 |- ( P e. Word (Vtx ` G ) -> P e. Fin )
33		hashnncl 13157	. . . . . . . . . . . . . . . . 17 |- ( P e. Fin -> ( ( # ` P ) e. NN <-> P =/= (/) ) )
34	33	bicomd 213	. . . . . . . . . . . . . . . 16 |- ( P e. Fin -> ( P =/= (/) <-> ( # ` P ) e. NN ) )
35	32, 34	syl 17	. . . . . . . . . . . . . . 15 |- ( P e. Word (Vtx ` G ) -> ( P =/= (/) <-> ( # ` P ) e. NN ) )
36	35	biimpac 503	. . . . . . . . . . . . . 14 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) -> ( # ` P ) e. NN )
37		wrdf 13310	. . . . . . . . . . . . . . . 16 |- ( P e. Word (Vtx ` G ) -> P : ( 0..^ ( # ` P ) ) --> (Vtx ` G ) )
38		nnz 11399	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` P ) e. NN -> ( # ` P ) e. ZZ )
39		fzoval 12471	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` P ) e. ZZ -> ( 0..^ ( # ` P ) ) = ( 0 ... ( ( # ` P ) - 1 ) ) )
40	38, 39	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` P ) e. NN -> ( 0..^ ( # ` P ) ) = ( 0 ... ( ( # ` P ) - 1 ) ) )
41	40	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( # ` P ) e. NN /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) -> ( 0..^ ( # ` P ) ) = ( 0 ... ( ( # ` P ) - 1 ) ) )
42		nnm1nn0 11334	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( # ` P ) e. NN -> ( ( # ` P ) - 1 ) e. NN0 )
43		fnfzo0hash 13234	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( # ` P ) - 1 ) e. NN0 /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) -> ( # ` f ) = ( ( # ` P ) - 1 ) )
44	42, 43	sylan 488	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( # ` P ) e. NN /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) -> ( # ` f ) = ( ( # ` P ) - 1 ) )
45	44	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( # ` P ) e. NN /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) -> ( ( # ` P ) - 1 ) = ( # ` f ) )
46	45	oveq2d 6666	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( # ` P ) e. NN /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) -> ( 0 ... ( ( # ` P ) - 1 ) ) = ( 0 ... ( # ` f ) ) )
47	41, 46	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` P ) e. NN /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) -> ( 0..^ ( # ` P ) ) = ( 0 ... ( # ` f ) ) )
48	47	feq2d 6031	. . . . . . . . . . . . . . . . . 18 |- ( ( ( # ` P ) e. NN /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) -> ( P : ( 0..^ ( # ` P ) ) --> (Vtx ` G ) <-> P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) ) )
49	48	biimpcd 239	. . . . . . . . . . . . . . . . 17 |- ( P : ( 0..^ ( # ` P ) ) --> (Vtx ` G ) -> ( ( ( # ` P ) e. NN /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) -> P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) ) )
50	49	expd 452	. . . . . . . . . . . . . . . 16 |- ( P : ( 0..^ ( # ` P ) ) --> (Vtx ` G ) -> ( ( # ` P ) e. NN -> ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) -> P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) ) ) )
51	37, 50	syl 17	. . . . . . . . . . . . . . 15 |- ( P e. Word (Vtx ` G ) -> ( ( # ` P ) e. NN -> ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) -> P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) ) ) )
52	51	adantl 482	. . . . . . . . . . . . . 14 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) -> ( ( # ` P ) e. NN -> ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) -> P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) ) ) )
53	36, 52	mpd 15	. . . . . . . . . . . . 13 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) -> ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) -> P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) ) )
54	53	3adant3 1081	. . . . . . . . . . . 12 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) -> P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) ) )
55	54	adantl 482	. . . . . . . . . . 11 |- ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) -> P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) ) )
56	55	com12 32	. . . . . . . . . 10 |- ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) -> ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) ) )
57	56	adantr 481	. . . . . . . . 9 |- ( ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) ) )
58	57	impcom 446	. . . . . . . 8 |- ( ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> P : ( 0 ... ( # ` f ) ) --> (Vtx ` G ) )
59		simpr 477	. . . . . . . . . 10 |- ( ( ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
60	36, 44	sylan 488	. . . . . . . . . . . . . . . . 17 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) -> ( # ` f ) = ( ( # ` P ) - 1 ) )
61	60	oveq2d 6666	. . . . . . . . . . . . . . . 16 |- ( ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) -> ( 0..^ ( # ` f ) ) = ( 0..^ ( ( # ` P ) - 1 ) ) )
62	61	ex 450	. . . . . . . . . . . . . . 15 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) ) -> ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) -> ( 0..^ ( # ` f ) ) = ( 0..^ ( ( # ` P ) - 1 ) ) ) )
63	62	3adant3 1081	. . . . . . . . . . . . . 14 |- ( ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) -> ( 0..^ ( # ` f ) ) = ( 0..^ ( ( # ` P ) - 1 ) ) ) )
64	63	adantl 482	. . . . . . . . . . . . 13 |- ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) -> ( 0..^ ( # ` f ) ) = ( 0..^ ( ( # ` P ) - 1 ) ) ) )
65	64	imp 445	. . . . . . . . . . . 12 |- ( ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) -> ( 0..^ ( # ` f ) ) = ( 0..^ ( ( # ` P ) - 1 ) ) )
66	65	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( 0..^ ( # ` f ) ) = ( 0..^ ( ( # ` P ) - 1 ) ) )
67	66	raleqdv 3144	. . . . . . . . . 10 |- ( ( ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
68	59, 67	mpbird 247	. . . . . . . . 9 |- ( ( ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
69	68	anasss 679	. . . . . . . 8 |- ( ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> A. i e. ( 0..^ ( # ` f ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
70	31, 58, 69	3jca 1242	. . . . . . 7 |- ( ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) /\ ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 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 ) ) } ) )
71	70	ex 450	. . . . . 6 |- ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> ( ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 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 ) ) } ) ) )
72	71	eximdv 1846	. . . . 5 |- ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> ( E. f ( f : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom (iEdg ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( (iEdg ` G ) ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> 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 ) ) } ) ) )
73	28, 72	mpd 15	. . . 4 |- ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ 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 ) ) } ) )
74	1, 7	upgriswlk 26537	. . . . . 6 |- ( 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 ) ) } ) ) )
75	74	adantr 481	. . . . 5 |- ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> ( 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 ) ) } ) ) )
76	75	exbidv 1850	. . . 4 |- ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> ( 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 ) ) } ) ) )
77	73, 76	mpbird 247	. . 3 |- ( ( G e. UPGraph /\ ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) -> E. f f (Walks ` G ) P )
78	77	ex 450	. 2 |- ( G e. UPGraph -> ( ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) -> E. f f (Walks ` G ) P ) )
79	3, 78	syl5bi 232	1 |- ( G e. UPGraph -> ( 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   E.wrex 2913   (/)c0 3915   {cpr 4179   class class class wbr 4653   dom cdm 5114   ran crn 5115   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   UPGraph cupgr 25975  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-ac2 9285  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-se 5074  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-isom 5897  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-ac 8939  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-wlks 26495  df-wwlks 26722

This theorem is referenced by:  wlkiswwlkupgr  26764  wlklnwwlklnupgr2  26771