Theorem uspgr2wlkeq 16076

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 1013	. . 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 16065	. . . 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 1227	. . 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 1039	. 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 10433	. . . . . . . . . . . 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 5627	. . . . . . . . . . . . 13 |- ( y = ( x + 1 ) -> ( ( 2nd ` A ) ` y ) = ( ( 2nd ` A ) ` ( x + 1 ) ) )
9		fveq2 5627	. . . . . . . . . . . . 13 |- ( y = ( x + 1 ) -> ( ( 2nd ` B ) ` y ) = ( ( 2nd ` B ) ` ( x + 1 ) ) )
10	8, 9	eqeq12d 2244	. . . . . . . . . . . 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 2910	. . . . . . . . . 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 2602	. . . . . . . 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 6024	. . . . . . . . . 10 |- ( y = x -> ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` A ) ` ( x + 1 ) ) )
16		fvoveq1 6024	. . . . . . . . . 10 |- ( y = x -> ( ( 2nd ` B ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) )
17	15, 16	eqeq12d 2244	. . . . . . . . 9 |- ( y = x -> ( ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) <-> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) )
18	17	cbvralvw 2769	. . . . . . . 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 10362	. . . . . . . . . 10 |- ( 0..^ N ) C_ ( 0 ... N )
21		ssralv 3288	. . . . . . . . . 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 2657	. . . . . . . . . . 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 3745	. . . . . . . . . . . . 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 2598	. . . . . . . . . . 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 15979	. . . . . . . 8 |- ( G e. USPGraph -> G e. UPGraph )
34		eqid 2229	. . . . . . . . . 10 |- (Vtx ` G ) = (Vtx ` G )
35		eqid 2229	. . . . . . . . . 10 |- (iEdg ` G ) = (iEdg ` G )
36		eqid 2229	. . . . . . . . . 10 |- ( 1st ` A ) = ( 1st ` A )
37		eqid 2229	. . . . . . . . . 10 |- ( 2nd ` A ) = ( 2nd ` A )
38	34, 35, 36, 37	upgrwlkcompim 16073	. . . . . . . . 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 2229	. . . . . . . . . 10 |- ( 1st ` B ) = ( 1st ` B )
42		eqid 2229	. . . . . . . . . 10 |- ( 2nd ` B ) = ( 2nd ` B )
43	34, 35, 41, 42	upgrwlkcompim 16073	. . . . . . . . 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 6009	. . . . . . . . . . . . . . . . . . 19 |- ( (♯ ` ( 1st ` B ) ) = N -> ( 0..^ (♯ ` ( 1st ` B ) ) ) = ( 0..^ N ) )
47	46	eqcoms 2232	. . . . . . . . . . . . . . . . . 18 |- ( N = (♯ ` ( 1st ` B ) ) -> ( 0..^ (♯ ` ( 1st ` B ) ) ) = ( 0..^ N ) )
48	47	raleqdv 2734	. . . . . . . . . . . . . . . . 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 6009	. . . . . . . . . . . . . . . . . . 19 |- ( (♯ ` ( 1st ` A ) ) = N -> ( 0..^ (♯ ` ( 1st ` A ) ) ) = ( 0..^ N ) )
50	49	eqcoms 2232	. . . . . . . . . . . . . . . . . 18 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0..^ (♯ ` ( 1st ` A ) ) ) = ( 0..^ N ) )
51	50	raleqdv 2734	. . . . . . . . . . . . . . . . 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 609	. . . . . . . . . . . . . . . 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 2657	. . . . . . . . . . . . . . . . 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 2239	. . . . . . . . . . . . . . . . . . . . 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 2239	. . . . . . . . . . . . . . . . . . . . . . 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 2232	. . . . . . . . . . . . . . . . . . . . . 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 2595	. . . . . . . . . . . . . . . . 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 1044	. . . . . . . . . . . 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 1044	. . . . . . . . . 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 1244	. . . . 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 2231	. . . . . . 7 |- ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) <-> ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) )
75	35	uspgrf1oedg 15974	. . . . . . . . . . . 12 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-onto-> (Edg ` G ) )
76		f1of1 5571	. . . . . . . . . . . 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 2230	. . . . . . . . . . . 12 |- ( G e. USPGraph -> (iEdg ` G ) = (iEdg ` G ) )
79		eqidd 2230	. . . . . . . . . . . 12 |- ( G e. USPGraph -> dom (iEdg ` G ) = dom (iEdg ` G ) )
80		edgvalg 15860	. . . . . . . . . . . . 13 |- ( G e. USPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
81	80	eqcomd 2235	. . . . . . . . . . . 12 |- ( G e. USPGraph -> ran (iEdg ` G ) = (Edg ` G ) )
82	78, 79, 81	f1eq123d 5564	. . . . . . . . . . 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 1042	. . . . . . . . 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 16064	. . . . . . . . . . . . . . 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 16064	. . . . . . . . . . . . . . 15 |- ( B e. (Walks ` G ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) )
88		oveq2 6009	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0..^ N ) = ( 0..^ (♯ ` ( 1st ` A ) ) ) )
89	88	eleq2d 2299	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N = (♯ ` ( 1st ` A ) ) -> ( y e. ( 0..^ N ) <-> y e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ) )
90		wrdsymbcl 11085	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6009	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N = (♯ ` ( 1st ` B ) ) -> ( 0..^ N ) = ( 0..^ (♯ ` ( 1st ` B ) ) ) )
98	97	eleq2d 2299	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N = (♯ ` ( 1st ` B ) ) -> ( y e. ( 0..^ N ) <-> y e. ( 0..^ (♯ ` ( 1st ` B ) ) ) ) )
99		wrdsymbcl 11085	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1039	. . . . . . . . . 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 5893	. . . . . . . 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 597	. . . . . . 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 2597	. . . . 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   {cpr 3667   dom cdm 4719   ran crn 4720   -->wf 5314   -1-1->wf1 5315   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6001   1stc1st 6284   2ndc2nd 6285   0cc0 7999   1c1 8000   + caddc 8002   ...cfz 10204  ..^cfzo 10338  ♯chash 10997  Word cword 11071  Vtxcvtx 15813  iEdgciedg 15814  Edgcedg 15858  UPGraphcupgr 15891  USPGraphcuspgr 15951  Walkscwlks 16030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-2o 6563  df-er 6680  df-map 6797  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-z 9447  df-dec 9579  df-uz 9723  df-fz 10205  df-fzo 10339  df-ihash 10998  df-word 11072  df-ndx 13035  df-slot 13036  df-base 13038  df-edgf 15806  df-vtx 15815  df-iedg 15816  df-edg 15859  df-uhgrm 15869  df-upgren 15893  df-uspgren 15953  df-wlks 16031

This theorem is referenced by:  uspgr2wlkeq2  16077