Theorem wlkeq 16509

Description: Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.)

Assertion

Ref	Expression
wlkeq	|- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )

Distinct variable groups:   x, A   x, B   x, N

Allowed substitution hint:   G( x)

Proof of Theorem wlkeq

Step	Hyp	Ref	Expression
1		eqid 2238	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2238	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
3		eqid 2238	. . . . . . 7 |- ( 1st ` A ) = ( 1st ` A )
4		eqid 2238	. . . . . . 7 |- ( 2nd ` A ) = ( 2nd ` A )
5	1, 2, 3, 4	wlkelwrd 16508	. . . . . 6 |- ( A e. (Walks ` G ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) )
6		eqid 2238	. . . . . . 7 |- ( 1st ` B ) = ( 1st ` B )
7		eqid 2238	. . . . . . 7 |- ( 2nd ` B ) = ( 2nd ` B )
8	1, 2, 6, 7	wlkelwrd 16508	. . . . . 6 |- ( B e. (Walks ` G ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) )
9	5, 8	anim12i 338	. . . . 5 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( ( ( 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 ) ) ) )
10		wlkmex 16474	. . . . . . 7 |- ( A e. (Walks ` G ) -> G e. _V )
11		wlkcprim 16505	. . . . . . 7 |- ( A e. (Walks ` G ) -> ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) )
12		wlklenvm1g 16497	. . . . . . 7 |- ( ( G e. _V /\ ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) ) -> (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) )
13	10, 11, 12	syl2anc 415	. . . . . 6 |- ( A e. (Walks ` G ) -> (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) )
14		wlkmex 16474	. . . . . . 7 |- ( B e. (Walks ` G ) -> G e. _V )
15		wlkcprim 16505	. . . . . . 7 |- ( B e. (Walks ` G ) -> ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) )
16		wlklenvm1g 16497	. . . . . . 7 |- ( ( G e. _V /\ ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) ) -> (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
17	14, 15, 16	syl2anc 415	. . . . . 6 |- ( B e. (Walks ` G ) -> (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
18	13, 17	anim12i 338	. . . . 5 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) /\ (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) ) )
19		eqwrd 11323	. . . . . . . 8 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 1st ` B ) e. Word dom (iEdg ` G ) ) -> ( ( 1st ` A ) = ( 1st ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
20	19	ad2ant2r 513	. . . . . . 7 |- ( ( ( ( 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 ) ) ) -> ( ( 1st ` A ) = ( 1st ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
21	20	adantr 276	. . . . . 6 |- ( ( ( ( ( 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 ) ) ) /\ ( (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) /\ (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) ) ) -> ( ( 1st ` A ) = ( 1st ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
22		lencl 11286	. . . . . . . . 9 |- ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> (♯ ` ( 1st ` A ) ) e. NN0 )
23	22	adantr 276	. . . . . . . 8 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) -> (♯ ` ( 1st ` A ) ) e. NN0 )
24		simpr 110	. . . . . . . 8 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) -> ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) )
25		simpr 110	. . . . . . . 8 |- ( ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) -> ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) )
26		2ffzeq 10526	. . . . . . . 8 |- ( ( (♯ ` ( 1st ` A ) ) e. NN0 /\ ( 2nd ` A ) : ( 0 ... (♯ ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ ( 2nd ` B ) : ( 0 ... (♯ ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
27	23, 24, 25, 26	syl2an3an 1339	. . . . . . 7 |- ( ( ( ( 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 ) ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
28	27	adantr 276	. . . . . 6 |- ( ( ( ( ( 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 ) ) ) /\ ( (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) /\ (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
29	21, 28	anbi12d 477	. . . . 5 |- ( ( ( ( ( 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 ) ) ) /\ ( (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) /\ (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
30	9, 18, 29	syl2anc 415	. . . 4 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
31	30	3adant3 1048	. . 3 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
32		eqeq1 2245	. . . . . . 7 |- ( N = (♯ ` ( 1st ` A ) ) -> ( N = (♯ ` ( 1st ` B ) ) <-> (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) ) )
33		oveq2 6083	. . . . . . . 8 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0..^ N ) = ( 0..^ (♯ ` ( 1st ` A ) ) ) )
34	33	raleqdv 2755	. . . . . . 7 |- ( N = (♯ ` ( 1st ` A ) ) -> ( A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) <-> A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) )
35	32, 34	anbi12d 477	. . . . . 6 |- ( N = (♯ ` ( 1st ` A ) ) -> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
36		oveq2 6083	. . . . . . . 8 |- ( N = (♯ ` ( 1st ` A ) ) -> ( 0 ... N ) = ( 0 ... (♯ ` ( 1st ` A ) ) ) )
37	36	raleqdv 2755	. . . . . . 7 |- ( N = (♯ ` ( 1st ` A ) ) -> ( A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) <-> A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) )
38	32, 37	anbi12d 477	. . . . . 6 |- ( N = (♯ ` ( 1st ` A ) ) -> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) <-> ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
39	35, 38	anbi12d 477	. . . . 5 |- ( N = (♯ ` ( 1st ` A ) ) -> ( ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
40	39	bibi2d 232	. . . 4 |- ( N = (♯ ` ( 1st ` A ) ) -> ( ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) <-> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) ) )
41	40	3ad2ant3 1051	. . 3 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) <-> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ (♯ ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... (♯ ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) ) )
42	31, 41	mpbird 167	. 2 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
43		wlkelvv 16504	. . . 4 |- ( A e. (Walks ` G ) -> A e. ( _V X. _V ) )
44		wlkelvv 16504	. . . 4 |- ( B e. (Walks ` G ) -> B e. ( _V X. _V ) )
45		xpopth 6400	. . . 4 |- ( ( A e. ( _V X. _V ) /\ B e. ( _V X. _V ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) )
46	43, 44, 45	syl2an 289	. . 3 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) )
47	46	3adant3 1048	. 2 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) )
48		3anass 1013	. . . 4 |- ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) <-> ( N = (♯ ` ( 1st ` B ) ) /\ ( A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
49		anandi 598	. . . 4 |- ( ( N = (♯ ` ( 1st ` B ) ) /\ ( A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
50	48, 49	bitr2i 185	. . 3 |- ( ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) )
51	50	a1i 9	. 2 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( ( ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
52	42, 47, 51	3bitr3d 218	1 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   class class class wbr 4125   X. cxp 4767   dom cdm 4769   -->wf 5368   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   0cc0 8169   1c1 8170   - cmin 8487   NN0cn0 9542   ...cfz 10390  ..^cfzo 10527  ♯chash 11192  Word cword 11282  Vtxcvtx 16167  iEdgciedg 16168  Walkscwlks 16472

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-map 6914  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-z 9624  df-dec 9757  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-wlks 16473

This theorem is referenced by:  uspgr2wlkeq  16520