Theorem uspgr2wlkeq 16606

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 1020	. . 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 16595	. . . 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 1234	. . 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 1046	. 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 10645	. . . . . . . . . . . 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 5695	. . . . . . . . . . . . 13 |- ( y = ( x + 1 ) -> ( ( 2nd ` A ) ` y ) = ( ( 2nd ` A ) ` ( x + 1 ) ) )
9		fveq2 5695	. . . . . . . . . . . . 13 |- ( y = ( x + 1 ) -> ( ( 2nd ` B ) ` y ) = ( ( 2nd ` B ) ` ( x + 1 ) ) )
10	8, 9	eqeq12d 2253	. . . . . . . . . . . 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 2932	. . . . . . . . . 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 2622	. . . . . . . 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 6108	. . . . . . . . . 10 |- ( y = x -> ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` A ) ` ( x + 1 ) ) )
16		fvoveq1 6108	. . . . . . . . . 10 |- ( y = x -> ( ( 2nd ` B ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) )
17	15, 16	eqeq12d 2253	. . . . . . . . 9 |- ( y = x -> ( ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) <-> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) )
18	17	cbvralvw 2790	. . . . . . . 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 10573	. . . . . . . . . 10 |- ( 0..^ N ) C_ ( 0 ... N )
21		ssralv 3312	. . . . . . . . . 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 2677	. . . . . . . . . . 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 3790	. . . . . . . . . . . . 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 2618	. . . . . . . . . . 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 16422	. . . . . . . 8 |- ( G e. USPGraph -> G e. UPGraph )
34		eqid 2238	. . . . . . . . . 10 |- (Vtx ` G ) = (Vtx ` G )
35		eqid 2238	. . . . . . . . . 10 |- (iEdg ` G ) = (iEdg ` G )
36		eqid 2238	. . . . . . . . . 10 |- ( 1st ` A ) = ( 1st ` A )
37		eqid 2238	. . . . . . . . . 10 |- ( 2nd ` A ) = ( 2nd ` A )
38	34, 35, 36, 37	upgrwlkcompim 16603	. . . . . . . . 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 2238	. . . . . . . . . 10 |- ( 1st ` B ) = ( 1st ` B )
42		eqid 2238	. . . . . . . . . 10 |- ( 2nd ` B ) = ( 2nd ` B )
43	34, 35, 41, 42	upgrwlkcompim 16603	. . . . . . . . 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 6093	. . . . . . . . . . . . . . . . . . 19 |- ( (♯ ` ( 1st ` B ) ) = N -> ( 0..^ (♯ ` ( 1st ` B ) ) ) = ( 0..^ N ) )
47	46	eqcoms 2241	. . . . . . . . . . . . . . . . . 18 |- ( N = (♯ ` ( 1st ` B ) ) -> ( 0..^ (♯ ` ( 1st ` B ) ) ) = ( 0..^ N ) )
48	47	raleqdv 2755	. . . . . . . . . . . . . . . . 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 6093	. . . . . . . . . . . . . . . . . . 19 |- ( (♯ ` ( 1st ` A ) ) = N -> ( 0..^ (♯ ` ( 1st ` A ) ) ) = ( 0..^ N ) )
50	49	eqcoms 2241	. . . . . . . . . . . . . . . . . 18 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0..^ (♯ ` ( 1st ` A ) ) ) = ( 0..^ N ) )
51	50	raleqdv 2755	. . . . . . . . . . . . . . . . 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 615	. . . . . . . . . . . . . . . 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 2677	. . . . . . . . . . . . . . . . 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 2248	. . . . . . . . . . . . . . . . . . . . 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 2248	. . . . . . . . . . . . . . . . . . . . . . 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 2241	. . . . . . . . . . . . . . . . . . . . . 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 2615	. . . . . . . . . . . . . . . . 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 1051	. . . . . . . . . . . 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 1051	. . . . . . . . . 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 1251	. . . . 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 2240	. . . . . . 7 |- ( ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) <-> ( (iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = ( (iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) )
75	35	uspgrf1oedg 16417	. . . . . . . . . . . 12 |- ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-onto-> (Edg ` G ) )
76		f1of1 5638	. . . . . . . . . . . 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 2239	. . . . . . . . . . . 12 |- ( G e. USPGraph -> (iEdg ` G ) = (iEdg ` G ) )
79		eqidd 2239	. . . . . . . . . . . 12 |- ( G e. USPGraph -> dom (iEdg ` G ) = dom (iEdg ` G ) )
80		edgvalg 16300	. . . . . . . . . . . . 13 |- ( G e. USPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
81	80	eqcomd 2244	. . . . . . . . . . . 12 |- ( G e. USPGraph -> ran (iEdg ` G ) = (Edg ` G ) )
82	78, 79, 81	f1eq123d 5631	. . . . . . . . . . 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 1049	. . . . . . . . 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 16594	. . . . . . . . . . . . . . 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 16594	. . . . . . . . . . . . . . 15 |- ( B e. (Walks ` G ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) )
88		oveq2 6093	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0..^ N ) = ( 0..^ (♯ ` ( 1st ` A ) ) ) )
89	88	eleq2d 2308	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N = (♯ ` ( 1st ` A ) ) -> ( y e. ( 0..^ N ) <-> y e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ) )
90		wrdsymbcl 11318	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6093	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N = (♯ ` ( 1st ` B ) ) -> ( 0..^ N ) = ( 0..^ (♯ ` ( 1st ` B ) ) ) )
98	97	eleq2d 2308	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N = (♯ ` ( 1st ` B ) ) -> ( y e. ( 0..^ N ) <-> y e. ( 0..^ (♯ ` ( 1st ` B ) ) ) ) )
99		wrdsymbcl 11318	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1046	. . . . . . . . . 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 5975	. . . . . . . 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 603	. . . . . . 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 2617	. . . . 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 397	. 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 ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   {cpr 3710   dom cdm 4774   ran crn 4775   -->wf 5373   -1-1->wf1 5374   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   0cc0 8179   1c1 8180   + caddc 8182   ...cfz 10411  ..^cfzo 10549  ♯chash 11214  Word cword 11304  Vtxcvtx 16253  iEdgciedg 16254  Edgcedg 16298  UPGraphcupgr 16332  USPGraphcuspgr 16394  Walkscwlks 16558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-uhgrm 16310  df-upgren 16334  df-uspgren 16396  df-wlks 16559

This theorem is used by:  uspgr2wlkeq2  16607