Theorem uspgr2wlkeq 16306

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 1016	. . 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 9	. 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 16295	. . . 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 1230	. . 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 1042	. 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 10535	. . . . . . . . . . . 12 |- ( x e. ( 0..^ N ) -> ( x + 1 ) e. ( 0 ... N ) )
7	6	adantl 277	. . . . . . . . . . 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 5648	. . . . . . . . . . . . 13 |- ( y = ( x + 1 ) -> ( ( 2nd ` A ) ` y ) = ( ( 2nd ` A ) ` ( x + 1 ) ) )
9		fveq2 5648	. . . . . . . . . . . . 13 |- ( y = ( x + 1 ) -> ( ( 2nd ` B ) ` y ) = ( ( 2nd ` B ) ` ( x + 1 ) ) )
10	8, 9	eqeq12d 2246	. . . . . . . . . . . 12 |- ( y = ( x + 1 ) -> ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) <-> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) )
11	10	adantl 277	. . . . . . . . . . 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 2914	. . . . . . . . . 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 260	. . . . . . . . 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 2605	. . . . . . . 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		fvoveq1 6051	. . . . . . . . . 10 |- ( y = x -> ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` A ) ` ( x + 1 ) ) )
16		fvoveq1 6051	. . . . . . . . . 10 |- ( y = x -> ( ( 2nd ` B ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) )
17	15, 16	eqeq12d 2246	. . . . . . . . 9 |- ( y = x -> ( ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) <-> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) )
18	17	cbvralvw 2772	. . . . . . . 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 ) ) )
19	14, 18	sylibr 134	. . . . . . 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 ) ) )
20		fzossfz 10463	. . . . . . . . . 10 |- ( 0..^ N ) C_ ( 0 ... N )
21		ssralv 3292	. . . . . . . . . 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 ) ) )
22	20, 21	mp1i 10	. . . . . . . . 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 ) ) )
23		r19.26 2660	. . . . . . . . . . 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 ) ) ) )
24		preq12 3754	. . . . . . . . . . . . 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 ) ) } )
25	24	a1i 9	. . . . . . . . . . . 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 ) ) } ) )
26	25	ralimdv 2601	. . . . . . . . . . 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 ) ) } ) )
27	23, 26	biimtrrid 153	. . . . . . . . . 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 ) ) } ) )
28	27	expd 258	. . . . . . . . 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 ) ) } ) ) )
29	22, 28	syld 45	. . . . . . . 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 ) ) } ) ) )
30	29	imp 124	. . . . . . 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 ) ) } ) )
31	19, 30	mpd 13	. . . . . 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 ) ) } )
32	31	ex 115	. . . . 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 ) ) } ) )
33		uspgrupgr 16122	. . . . . . . 8 |- ( G e. USPGraph -> G e. UPGraph )
34		eqid 2231	. . . . . . . . . 10 |- (Vtx ` G ) = (Vtx ` G )
35		eqid 2231	. . . . . . . . . 10 |- (iEdg ` G ) = (iEdg ` G )
36		eqid 2231	. . . . . . . . . 10 |- ( 1st ` A ) = ( 1st ` A )
37		eqid 2231	. . . . . . . . . 10 |- ( 2nd ` A ) = ( 2nd ` A )
38	34, 35, 36, 37	upgrwlkcompim 16303	. . . . . . . . 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 ) ) } ) )
39	38	ex 115	. . . . . . . 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 ) ) } ) ) )
40	33, 39	syl 14	. . . . . . 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 ) ) } ) ) )
41		eqid 2231	. . . . . . . . . 10 |- ( 1st ` B ) = ( 1st ` B )
42		eqid 2231	. . . . . . . . . 10 |- ( 2nd ` B ) = ( 2nd ` B )
43	34, 35, 41, 42	upgrwlkcompim 16303	. . . . . . . . 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 ) ) } ) )
44	43	ex 115	. . . . . . . 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 ) ) } ) ) )
45	33, 44	syl 14	. . . . . . 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 ) ) } ) ) )
46		oveq2 6036	. . . . . . . . . . . . . . . . . . 19 |- ( (♯ ` ( 1st ` B ) ) = N -> ( 0..^ (♯ ` ( 1st ` B ) ) ) = ( 0..^ N ) )
47	46	eqcoms 2234	. . . . . . . . . . . . . . . . . 18 |- ( N = (♯ ` ( 1st ` B ) ) -> ( 0..^ (♯ ` ( 1st ` B ) ) ) = ( 0..^ N ) )
48	47	raleqdv 2737	. . . . . . . . . . . . . . . . 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 ) ) } ) )
49		oveq2 6036	. . . . . . . . . . . . . . . . . . 19 |- ( (♯ ` ( 1st ` A ) ) = N -> ( 0..^ (♯ ` ( 1st ` A ) ) ) = ( 0..^ N ) )
50	49	eqcoms 2234	. . . . . . . . . . . . . . . . . 18 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0..^ (♯ ` ( 1st ` A ) ) ) = ( 0..^ N ) )
51	50	raleqdv 2737	. . . . . . . . . . . . . . . . 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 ) ) } ) )
52	48, 51	bi2anan9r 611	. . . . . . . . . . . . . . . 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 ) ) } ) ) )
53		r19.26 2660	. . . . . . . . . . . . . . . . 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 ) ) } ) )
54		eqeq2 2241	. . . . . . . . . . . . . . . . . . . . 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 ) ) } ) )
55		eqeq2 2241	. . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
56	55	eqcoms 2234	. . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
57	56	biimpd 144	. . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
58	54, 57	biimtrdi 163	. . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
59	58	com13 80	. . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
60	59	imp 124	. . . . . . . . . . . . . . . . . 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 ) ) ) )
61	60	ral2imi 2598	. . . . . . . . . . . . . . . . 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 ) ) ) )
62	53, 61	sylbir 135	. . . . . . . . . . . . . . . 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 ) ) ) )
63	52, 62	biimtrdi 163	. . . . . . . . . . . . . . 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 ) ) ) ) )
64	63	com12 30	. . . . . . . . . . . . . 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 ) ) ) ) )
65	64	ex 115	. . . . . . . . . . . . 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 ) ) ) ) ) )
66	65	3ad2ant3 1047	. . . . . . . . . . . 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 ) ) ) ) ) )
67	66	com12 30	. . . . . . . . . . 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 ) ) ) ) ) )
68	67	3ad2ant3 1047	. . . . . . . . . 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 ) ) ) ) ) )
69	68	imp 124	. . . . . . . . 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 ) ) ) ) )
70	69	expd 258	. . . . . . . 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 ) ) ) ) ) )
71	70	a1i 9	. . . . . . 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 ) ) ) ) ) ) )
72	40, 45, 71	syl2and 295	. . . . . 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 ) ) ) ) ) ) )
73	72	3imp1 1247	. . . . 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 ) ) ) )
74		eqcom 2233	. . . . . . 7 |- ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) <-> ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) )
75	35	uspgrf1oedg 16117	. . . . . . . . . . . 12 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-onto-> (Edg ` G ) )
76		f1of1 5591	. . . . . . . . . . . 12 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-onto-> (Edg ` G ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) )
77	75, 76	syl 14	. . . . . . . . . . 11 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) )
78		eqidd 2232	. . . . . . . . . . . 12 |- ( G e. USPGraph -> (iEdg ` G ) = (iEdg ` G ) )
79		eqidd 2232	. . . . . . . . . . . 12 |- ( G e. USPGraph -> dom (iEdg ` G ) = dom (iEdg ` G ) )
80		edgvalg 16000	. . . . . . . . . . . . 13 |- ( G e. USPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
81	80	eqcomd 2237	. . . . . . . . . . . 12 |- ( G e. USPGraph -> ran (iEdg ` G ) = (Edg ` G ) )
82	78, 79, 81	f1eq123d 5584	. . . . . . . . . . 11 |- ( G e. USPGraph -> ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> ran (iEdg ` G ) <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> (Edg ` G ) ) )
83	77, 82	mpbird 167	. . . . . . . . . 10 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> ran (iEdg ` G ) )
84	83	3ad2ant1 1045	. . . . . . . . 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 ) )
85	84	adantr 276	. . . . . . . 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 ) )
86	34, 35, 36, 37	wlkelwrd 16294	. . . . . . . . . . . . . . 15 |- ( A e. (Walks ` G ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) )
87	34, 35, 41, 42	wlkelwrd 16294	. . . . . . . . . . . . . . 15 |- ( B e. (Walks ` G ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) )
88		oveq2 6036	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0..^ N ) = ( 0..^ (♯ ` ( 1st ` A ) ) ) )
89	88	eleq2d 2301	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N = (♯ ` ( 1st ` A ) ) -> ( y e. ( 0..^ N ) <-> y e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ) )
90		wrdsymbcl 11193	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ y e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ) -> ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) )
91	90	expcom 116	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( y e. ( 0..^ (♯ ` ( 1st ` A ) ) ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) ) )
92	89, 91	biimtrdi 163	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( N = (♯ ` ( 1st ` A ) ) -> ( y e. ( 0..^ N ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> ( ( 1st ` A ) ` y ) e. dom (iEdg ` G ) ) ) )
93	92	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
94	93	imp 124	. . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
95	94	com12 30	. . . . . . . . . . . . . . . . . . . . . 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 ) ) )
96	95	adantl 277	. . . . . . . . . . . . . . . . . . . . 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 ) ) )
97		oveq2 6036	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N = (♯ ` ( 1st ` B ) ) -> ( 0..^ N ) = ( 0..^ (♯ ` ( 1st ` B ) ) ) )
98	97	eleq2d 2301	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N = (♯ ` ( 1st ` B ) ) -> ( y e. ( 0..^ N ) <-> y e. ( 0..^ (♯ ` ( 1st ` B ) ) ) ) )
99		wrdsymbcl 11193	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ y e. ( 0..^ (♯ ` ( 1st ` B ) ) ) ) -> ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) )
100	99	expcom 116	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( y e. ( 0..^ (♯ ` ( 1st ` B ) ) ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) -> ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) )
101	98, 100	biimtrdi 163	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( N = (♯ ` ( 1st ` B ) ) -> ( y e. ( 0..^ N ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) -> ( ( 1st ` B ) ` y ) e. dom (iEdg ` G ) ) ) )
102	101	adantl 277	. . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
103	102	imp 124	. . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
104	103	com12 30	. . . . . . . . . . . . . . . . . . . . . 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 ) ) )
105	104	adantr 276	. . . . . . . . . . . . . . . . . . . . 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 ) ) )
106	96, 105	jcad 307	. . . . . . . . . . . . . . . . . . . 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 ) ) ) )
107	106	ex 115	. . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
108	107	adantr 276	. . . . . . . . . . . . . . . . . 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 ) ) ) ) )
109	108	com12 30	. . . . . . . . . . . . . . . . 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 ) ) ) ) )
110	109	adantr 276	. . . . . . . . . . . . . . . 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 ) ) ) ) )
111	110	imp 124	. . . . . . . . . . . . . . 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 ) ) ) )
112	86, 87, 111	syl2an 289	. . . . . . . . . . . . . 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 ) ) ) )
113	112	expd 258	. . . . . . . . . . . . 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 ) ) ) ) )
114	113	expd 258	. . . . . . . . . . . 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 ) ) ) ) ) )
115	114	imp 124	. . . . . . . . . . 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 ) ) ) ) )
116	115	3adant1 1042	. . . . . . . . . 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 ) ) ) ) )
117	116	imp 124	. . . . . . . . 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 ) ) ) )
118	117	imp 124	. . . . . . . 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 ) ) )
119		f1veqaeq 5920	. . . . . . . 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 ) ) )
120	85, 118, 119	syl2an2r 599	. . . . . . 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 ) ) )
121	74, 120	biimtrid 152	. . . . . 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 ) ) )
122	121	ralimdva 2600	. . . . 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 ) ) )
123	32, 73, 122	3syld 57	. . . 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 ) ) )
124	123	expimpd 363	. . 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 ) ) )
125	124	pm4.71d 393	. 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 ) ) ) )
126	2, 5, 125	3bitr4d 220	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   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   {cpr 3674   dom cdm 4731   ran crn 4732   -->wf 5329   -1-1->wf1 5330   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   1stc1st 6310   2ndc2nd 6311   0cc0 8092   1c1 8093   + caddc 8095   ...cfz 10305  ..^cfzo 10439  ♯chash 11100  Word cword 11179  Vtxcvtx 15953  iEdgciedg 15954  Edgcedg 15998  UPGraphcupgr 16032  USPGraphcuspgr 16094  Walkscwlks 16258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-9 9268  df-n0 9462  df-z 9541  df-dec 9673  df-uz 9817  df-fz 10306  df-fzo 10440  df-ihash 11101  df-word 11180  df-ndx 13165  df-slot 13166  df-base 13168  df-edgf 15946  df-vtx 15955  df-iedg 15956  df-edg 15999  df-uhgrm 16010  df-upgren 16034  df-uspgren 16096  df-wlks 16259

This theorem is referenced by:  uspgr2wlkeq2  16307