Theorem uspgr2wlkeq 26542

Description: Conditions for two walks within the same simple pseudograph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.) (Revised by AV, 14-Apr-2021.)

Assertion

Ref	Expression
uspgr2wlkeq	|- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) )

Distinct variable groups:   y, A   y, B   y, G   y, N

Proof of Theorem uspgr2wlkeq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3anan32 1050	. . 3 |- ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) <-> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) /\ A. y e. ( 0..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) )
2	1	a1i 11	. 2 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) <-> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) /\ A. y e. ( 0..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) ) )
3		wlkeq 26529	. . . 4 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) )
4	3	3expa 1265	. . 3 |- ( ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) )
5	4	3adant1 1079	. 2 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) )
6		fzofzp1 12565	. . . . . . . . . . . 12 |- ( x e. ( 0..^ N ) -> ( x + 1 ) e. ( 0 ... N ) )
7	6	adantl 482	. . . . . . . . . . 11 |- ( ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ x e. ( 0..^ N ) ) -> ( x + 1 ) e. ( 0 ... N ) )
8		fveq2 6191	. . . . . . . . . . . . 13 |- ( y = ( x + 1 ) -> ( ( 2nd ` A ) ` y ) = ( ( 2nd ` A ) ` ( x + 1 ) ) )
9		fveq2 6191	. . . . . . . . . . . . 13 |- ( y = ( x + 1 ) -> ( ( 2nd ` B ) ` y ) = ( ( 2nd ` B ) ` ( x + 1 ) ) )
10	8, 9	eqeq12d 2637	. . . . . . . . . . . 12 |- ( y = ( x + 1 ) -> ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) <-> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) )
11	10	adantl 482	. . . . . . . . . . 11 |- ( ( ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ x e. ( 0..^ N ) ) /\ y = ( x + 1 ) ) -> ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) <-> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) )
12	7, 11	rspcdv 3312	. . . . . . . . . 10 |- ( ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ x e. ( 0..^ N ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) )
13	12	impancom 456	. . . . . . . . 9 |- ( ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> ( x e. ( 0..^ N ) -> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) )
14	13	ralrimiv 2965	. . . . . . . 8 |- ( ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> A. x e. ( 0..^ N ) ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) )
15		oveq1 6657	. . . . . . . . . . 11 |- ( y = x -> ( y + 1 ) = ( x + 1 ) )
16	15	fveq2d 6195	. . . . . . . . . 10 |- ( y = x -> ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` A ) ` ( x + 1 ) ) )
17	15	fveq2d 6195	. . . . . . . . . 10 |- ( y = x -> ( ( 2nd ` B ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) )
18	16, 17	eqeq12d 2637	. . . . . . . . 9 |- ( y = x -> ( ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) <-> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) )
19	18	cbvralv 3171	. . . . . . . 8 |- ( A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) <-> A. x e. ( 0..^ N ) ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) )
20	14, 19	sylibr 224	. . . . . . 7 |- ( ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) )
21		fzossfz 12488	. . . . . . . . . 10 |- ( 0..^ N ) C_ ( 0 ... N )
22		ssralv 3666	. . . . . . . . . 10 |- ( ( 0..^ N ) C_ ( 0 ... N ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) )
23	21, 22	mp1i 13	. . . . . . . . 9 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) )
24		r19.26 3064	. . . . . . . . . . 11 |- ( A. y e. ( 0..^ N ) ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) <-> ( A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) )
25		preq12 4270	. . . . . . . . . . . . 13 |- ( ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) -> { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } )
26	25	a1i 11	. . . . . . . . . . . 12 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) -> { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) )
27	26	ralimdv 2963	. . . . . . . . . . 11 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0..^ N ) ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) -> A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) )
28	24, 27	syl5bir 233	. . . . . . . . . 10 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( ( A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) -> A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) )
29	28	expd 452	. . . . . . . . 9 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> ( A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) -> A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) )
30	23, 29	syld 47	. . . . . . . 8 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> ( A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) -> A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) )
31	30	imp 445	. . . . . . 7 |- ( ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> ( A. y e. ( 0..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) -> A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) )
32	20, 31	mpd 15	. . . . . 6 |- ( ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } )
33	32	ex 450	. . . . 5 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) )
34		uspgrupgr 26071	. . . . . . . 8 |- ( G e. USPGraph -> G e. UPGraph )
35		eqid 2622	. . . . . . . . . 10 |- (Vtx ` G ) = (Vtx ` G )
36		eqid 2622	. . . . . . . . . 10 |- (iEdg ` G ) = (iEdg ` G )
37		eqid 2622	. . . . . . . . . 10 |- ( 1st ` A ) = ( 1st ` A )
38		eqid 2622	. . . . . . . . . 10 |- ( 2nd ` A ) = ( 2nd ` A )
39	35, 36, 37, 38	upgrwlkcompim 26539	. . . . . . . . 9 |- ( ( G e. UPGraph /\ A e. (Walks ` G ) ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) )
40	39	ex 450	. . . . . . . 8 |- ( G e. UPGraph -> ( A e. (Walks ` G ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) ) )
41	34, 40	syl 17	. . . . . . 7 |- ( G e. USPGraph -> ( A e. (Walks ` G ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) ) )
42		eqid 2622	. . . . . . . . . 10 |- ( 1st ` B ) = ( 1st ` B )
43		eqid 2622	. . . . . . . . . 10 |- ( 2nd ` B ) = ( 2nd ` B )
44	35, 36, 42, 43	upgrwlkcompim 26539	. . . . . . . . 9 |- ( ( G e. UPGraph /\ B e. (Walks ` G ) ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) )
45	44	ex 450	. . . . . . . 8 |- ( G e. UPGraph -> ( B e. (Walks ` G ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) )
46	34, 45	syl 17	. . . . . . 7 |- ( G e. USPGraph -> ( B e. (Walks ` G ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) )
47		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` ( 1st ` B ) ) = N -> ( 0..^ ( # ` ( 1st ` B ) ) ) = ( 0..^ N ) )
48	47	eqcoms 2630	. . . . . . . . . . . . . . . . . 18 |- ( N = ( # ` ( 1st ` B ) ) -> ( 0..^ ( # ` ( 1st ` B ) ) ) = ( 0..^ N ) )
49	48	raleqdv 3144	. . . . . . . . . . . . . . . . 17 |- ( N = ( # ` ( 1st ` B ) ) -> ( A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } <-> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) )
50		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` ( 1st ` A ) ) = N -> ( 0..^ ( # ` ( 1st ` A ) ) ) = ( 0..^ N ) )
51	50	eqcoms 2630	. . . . . . . . . . . . . . . . . 18 |- ( N = ( # ` ( 1st ` A ) ) -> ( 0..^ ( # ` ( 1st ` A ) ) ) = ( 0..^ N ) )
52	51	raleqdv 3144	. . . . . . . . . . . . . . . . 17 |- ( N = ( # ` ( 1st ` A ) ) -> ( A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } <-> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) )
53	49, 52	bi2anan9r 918	. . . . . . . . . . . . . . . 16 |- ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( ( A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) <-> ( A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) ) )
54		r19.26 3064	. . . . . . . . . . . . . . . . 17 |- ( A. y e. ( 0..^ N ) ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) <-> ( A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) )
55		eqeq2 2633	. . . . . . . . . . . . . . . . . . . . 21 |- ( { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } <-> ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) )
56		eqeq2 2633	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) -> ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } <-> ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) )
57	56	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } <-> ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) )
58	57	biimpd 219	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) )
59	55, 58	syl6bi 243	. . . . . . . . . . . . . . . . . . . 20 |- ( { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } -> ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) )
60	59	com13 88	. . . . . . . . . . . . . . . . . . 19 |- ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } -> ( { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) )
61	60	imp 445	. . . . . . . . . . . . . . . . . 18 |- ( ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) )
62	61	ral2imi 2947	. . . . . . . . . . . . . . . . 17 |- ( A. y e. ( 0..^ N ) ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) )
63	54, 62	sylbir 225	. . . . . . . . . . . . . . . 16 |- ( ( A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) )
64	53, 63	syl6bi 243	. . . . . . . . . . . . . . 15 |- ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( ( A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) )
65	64	com12 32	. . . . . . . . . . . . . 14 |- ( ( A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) )
66	65	ex 450	. . . . . . . . . . . . 13 |- ( A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) )
67	66	3ad2ant3 1084	. . . . . . . . . . . 12 |- ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) -> ( A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) )
68	67	com12 32	. . . . . . . . . . 11 |- ( A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } -> ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) )
69	68	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) )
70	69	imp 445	. . . . . . . . 9 |- ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) )
71	70	expd 452	. . . . . . . 8 |- ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) -> ( N = ( # ` ( 1st ` A ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) )
72	71	a1i 11	. . . . . . 7 |- ( G e. USPGraph -> ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) /\ A. y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) -> ( N = ( # ` ( 1st ` A ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) ) )
73	41, 46, 72	syl2and 500	. . . . . 6 |- ( G e. USPGraph -> ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( N = ( # ` ( 1st ` A ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) ) )
74	73	3imp1 1280	. . . . 5 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) )
75		eqcom 2629	. . . . . . 7 |- ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) <-> ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) )
76	36	uspgrf1oedg 26068	. . . . . . . . . . . 12 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-onto-> (Edg ` G ) )
77		f1of1 6136	. . . . . . . . . . . 12 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-onto-> (Edg ` G ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) )
78	76, 77	syl 17	. . . . . . . . . . 11 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) )
79		eqidd 2623	. . . . . . . . . . . 12 |- ( G e. USPGraph -> (iEdg ` G ) = (iEdg ` G ) )
80		eqidd 2623	. . . . . . . . . . . 12 |- ( G e. USPGraph -> dom (iEdg ` G ) = dom (iEdg ` G ) )
81		edgval 25941	. . . . . . . . . . . . . 14 |- (Edg ` G ) = ran (iEdg ` G )
82	81	eqcomi 2631	. . . . . . . . . . . . 13 |- ran (iEdg ` G ) = (Edg ` G )
83	82	a1i 11	. . . . . . . . . . . 12 |- ( G e. USPGraph -> ran (iEdg ` G ) = (Edg ` G ) )
84	79, 80, 83	f1eq123d 6131	. . . . . . . . . . 11 |- ( G e. USPGraph -> ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> ran (iEdg ` G ) <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) ) )
85	78, 84	mpbird 247	. . . . . . . . . 10 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> ran (iEdg ` G ) )
86	85	3ad2ant1 1082	. . . . . . . . 9 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> ran (iEdg ` G ) )
87	86	adantr 481	. . . . . . . 8 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> ran (iEdg ` G ) )
88	35, 36, 37, 38	wlkelwrd 26528	. . . . . . . . . . . . . . 15 |- ( A e. (Walks ` G ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) )
89	35, 36, 42, 43	wlkelwrd 26528	. . . . . . . . . . . . . . 15 |- ( B e. (Walks ` G ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) )
90		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N = ( # ` ( 1st ` A ) ) -> ( 0..^ N ) = ( 0..^ ( # ` ( 1st ` A ) ) ) )
91	90	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N = ( # ` ( 1st ` A ) ) -> ( y e. ( 0..^ N ) <-> y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ) )
92		wrdsymbcl 13318	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ y e. ( 0..^ ( # ` ( 1st ` A ) ) ) ) -> ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) )
93	92	expcom 451	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( y e. ( 0..^ ( # ` ( 1st ` A ) ) ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) ) )
94	91, 93	syl6bi 243	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( N = ( # ` ( 1st ` A ) ) -> ( y e. ( 0..^ N ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) ) ) )
95	94	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( y e. ( 0..^ N ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) ) ) )
96	95	imp 445	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) ) )
97	96	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) ) )
98	97	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 1st ` A ) e. Word dom (iEdg ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) ) )
99		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N = ( # ` ( 1st ` B ) ) -> ( 0..^ N ) = ( 0..^ ( # ` ( 1st ` B ) ) ) )
100	99	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N = ( # ` ( 1st ` B ) ) -> ( y e. ( 0..^ N ) <-> y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ) )
101		wrdsymbcl 13318	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ y e. ( 0..^ ( # ` ( 1st ` B ) ) ) ) -> ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) )
102	101	expcom 451	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( y e. ( 0..^ ( # ` ( 1st ` B ) ) ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) -> ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) )
103	100, 102	syl6bi 243	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( N = ( # ` ( 1st ` B ) ) -> ( y e. ( 0..^ N ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) -> ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) )
104	103	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( y e. ( 0..^ N ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) -> ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) )
105	104	imp 445	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) -> ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) )
106	105	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 1st ` B ) e. Word dom (iEdg ` G ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) )
107	106	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 1st ` A ) e. Word dom (iEdg ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) )
108	98, 107	jcad 555	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 1st ` A ) e. Word dom (iEdg ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) )
109	108	ex 450	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1st ` B ) e. Word dom (iEdg ` G ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) ) )
110	109	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) ) )
111	110	com12 32	. . . . . . . . . . . . . . . . 17 |- ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) ) )
112	111	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) -> ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) ) )
113	112	imp 445	. . . . . . . . . . . . . . 15 |- ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) )
114	88, 89, 113	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) )
115	114	expd 452	. . . . . . . . . . . . 13 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( y e. ( 0..^ N ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) ) )
116	115	expd 452	. . . . . . . . . . . 12 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( N = ( # ` ( 1st ` A ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( y e. ( 0..^ N ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) ) ) )
117	116	imp 445	. . . . . . . . . . 11 |- ( ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( y e. ( 0..^ N ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) ) )
118	117	3adant1 1079	. . . . . . . . . 10 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( y e. ( 0..^ N ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) ) )
119	118	imp 445	. . . . . . . . 9 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( y e. ( 0..^ N ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) )
120	119	imp 445	. . . . . . . 8 |- ( ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) )
121		f1veqaeq 6514	. . . . . . . 8 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> ran (iEdg ` G ) /\ ( ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) -> ( ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) -> ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) )
122	87, 120, 121	syl2an2r 876	. . . . . . 7 |- ( ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) -> ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) )
123	75, 122	syl5bi 232	. . . . . 6 |- ( ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0..^ N ) ) -> ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) -> ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) )
124	123	ralimdva 2962	. . . . 5 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0..^ N ) ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) -> A. y e. ( 0..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) )
125	33, 74, 124	3syld 60	. . . 4 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> A. y e. ( 0..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) )
126	125	expimpd 629	. . 3 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> A. y e. ( 0..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) )
127	126	pm4.71d 666	. 2 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) <-> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) /\ A. y e. ( 0..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) ) )
128	2, 5, 127	3bitr4d 300	1 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {cpr 4179   dom cdm 5114   ran crn 5115   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   1c1 9937   + caddc 9939   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UPGraph cupgr 25975   USPGraph cuspgr 26043  Walkscwlks 26492

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

This theorem is referenced by:  uspgr2wlkeq2  26543  clwlksf1clwwlk  26969